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","Browsers: MathJax works with all modern browsers including"," Edge, Firefox, Chrome, Safari, Opera, and most mobile browsers.
","Math Menu: MathJax adds a contextual menu to equations."," Right-click or CTRL-click on any mathematics to access the menu.
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Show Math As: These options allow you to view the formula's"," source markup (as MathML or in its original format).
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Copy to Clipboard: These options copy the formula's source markup,"," as MathML or in its original format, to the clipboard"," (in browsers that support that).
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","
Accessibility: MathJax can work with screen"," readers to make mathematics accessible to the visually impaired."," Turn on the explorer to enable generation of speech strings"," and the ability to investigate expressions interactively.
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","
","Math Zoom: If you are having difficulty reading an"," equation, MathJax can enlarge it to help you see it better, or"," you can scall all the math on the page to make it larger."," Turn these features on in the Math Settings menu.
","Preferences: MathJax uses your browser's localStorage database"," to save the preferences set via this menu locally in your browser. These"," are not used to track you, and are not transferred or used remotely by"," MathJax in any way.
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o=n.load(e).then((function(){t.loading--,t.loadingPromises.delete(e),r(),0===t.loading&&t._loadingPromise&&(t._loadingPromise=null,t._loadingOK())})).catch((function(e){t._loadingPromise?(t._loadingPromise=null,t._loadingFailed(e)):console.log(e)}));t.loadingPromises.set(e,o)}}},t.prototype.loadA11y=function(e){var r=this,n=!a.STATE.ENRICHED;this.loadComponent("a11y/"+e,(function(){var o=m.startup;i.mathjax.handlers.unregister(o.handler),o.handler=o.getHandler(),i.mathjax.handlers.register(o.handler);var s=r.document;r.document=o.document=o.getDocument(),r.document.menu=r,r.document.outputJax.reset(),r.transferMathList(s),r.document.processed=s.processed,t._loadingPromise||(r.document.outputJax.reset(),r.rerender("complexity"===e||n?a.STATE.COMPILED:a.STATE.TYPESET))}))},t.prototype.transferMathList=function(t){var e,r,n=this.document.options.MathItem;try{for(var i=o(t.math),a=i.next();!a.done;a=i.next()){var s=a.value,l=new 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e=this.store;this.moving||e.insertTaborder(),e.active.focus()}},e.prototype.left=function(t){this.move_(this.store.previous())},e.prototype.right=function(t){this.move_(this.store.next())},Object.defineProperty(e.prototype,"frame",{get:function(){return this._frame},enumerable:!1,configurable:!0}),Object.defineProperty(e.prototype,"store",{get:function(){return this._store},enumerable:!1,configurable:!0}),e.prototype.post=function(e,r){if(void 0!==r)return this.moving||this.store.removeTaborder(),void t.prototype.post.call(this,e,r);var n,o,i,a=e;if(a instanceof Event?(n=a.target,this.stop(a)):n=a,a instanceof MouseEvent&&(o=a.pageX,i=a.pageY,o||i||!a.clientX||(o=a.clientX+document.body.scrollLeft+document.documentElement.scrollLeft,i=a.clientY+document.body.scrollTop+document.documentElement.scrollTop)),!o&&!i&&n){var s=window.pageXOffset||document.documentElement.scrollLeft,l=window.pageYOffset||document.documentElement.scrollTop,c=n.getBoundingClientRect();o=(c.right+c.left)/2+s,i=(c.bottom+c.top)/2+l}this.store.active=n,this.anchor=this.store.active;var u=this.html;o+u.offsetWidth>document.body.offsetWidth-5&&(o=document.body.offsetWidth-u.offsetWidth-5),this.post(o,i)},e.prototype.registerWidget=function(t){this.widgets.push(t)},e.prototype.unregisterWidget=function(t){var e=this.widgets.indexOf(t);e>-1&&this.widgets.splice(e,1),0===this.widgets.length&&this.unpost()},e.prototype.unpostWidgets=function(){this.widgets.forEach((function(t){return t.unpost()}))},e.prototype.toJson=function(){return{type:""}},e.prototype.move_=function(t){this.anchor&&t!==this.anchor&&(this.moving=!0,this.unpost(),this.post(t),this.moving=!1)},e}(i.AbstractMenu);e.ContextMenu=c},7309:function(t,e,r){Object.defineProperty(e,"__esModule",{value:!0}),e.CssStyles=void 0;var n=r(2165);!function(t){function e(t){return"."+(n.HtmlClasses[t]||t)}var r={};r[e("INFOCLOSE")]="{ top:.2em; right:.2em;}",r[e("INFOCONTENT")]="{ overflow:auto; text-align:left; font-size:80%; padding:.4em .6em; border:1px inset; margin:1em 0px; max-height:20em; max-width:30em; background-color:#EEEEEE; white-space:normal;}",r[e("INFO")+e("MOUSEPOST")]="{outline:none;}",r[e("INFO")]='{ position:fixed; left:50%; width:auto; text-align:center; border:3px outset; padding:1em 2em; background-color:#DDDDDD; color:black; cursor:default; font-family:message-box; font-size:120%; font-style:normal; text-indent:0; text-transform:none; line-height:normal; letter-spacing:normal; word-spacing:normal; word-wrap:normal; white-space:nowrap; float:none; z-index:201; border-radius: 15px; /* Opera 10.5 and IE9 */ -webkit-border-radius:15px; /* Safari and Chrome */ -moz-border-radius:15px; /* Firefox */ -khtml-border-radius:15px; /* Konqueror */ box-shadow:0px 10px 20px #808080; /* Opera 10.5 and IE9 */ -webkit-box-shadow:0px 10px 20px #808080; /* Safari 3 & Chrome */ -moz-box-shadow:0px 10px 20px #808080; /* Forefox 3.5 */ -khtml-box-shadow:0px 10px 20px #808080; /* Konqueror */ filter:progid:DXImageTransform.Microsoft.dropshadow(OffX=2, OffY=2, Color="gray", Positive="true"); /* IE */}';var o={};o[e("MENU")]="{ position:absolute; background-color:white; color:black; width:auto; padding:5px 0px; border:1px solid #CCCCCC; margin:0; cursor:default; font: menu; text-align:left; text-indent:0; text-transform:none; line-height:normal; letter-spacing:normal; word-spacing:normal; word-wrap:normal; white-space:nowrap; float:none; z-index:201; border-radius: 5px; /* Opera 10.5 and IE9 */ -webkit-border-radius: 5px; /* Safari and Chrome */ -moz-border-radius: 5px; /* Firefox */ -khtml-border-radius: 5px; /* Konqueror */ box-shadow:0px 10px 20px #808080; /* Opera 10.5 and IE9 */ -webkit-box-shadow:0px 10px 20px #808080; /* Safari 3 & Chrome */ -moz-box-shadow:0px 10px 20px #808080; /* Forefox 3.5 */ -khtml-box-shadow:0px 10px 20px #808080; /* Konqueror */}",o[e("MENUITEM")]="{ padding: 1px 2em; background:transparent;}",o[e("MENUARROW")]="{ position:absolute; right:.5em; padding-top:.25em; color:#666666; font-family: null; font-size: .75em}",o[e("MENUACTIVE")+" "+e("MENUARROW")]="{color:white}",o[e("MENUARROW")+e("RTL")]="{left:.5em; right:auto}",o[e("MENUCHECK")]="{ position:absolute; left:.7em; font-family: null}",o[e("MENUCHECK")+e("RTL")]="{ right:.7em; left:auto }",o[e("MENURADIOCHECK")]="{ position:absolute; left: .7em;}",o[e("MENURADIOCHECK")+e("RTL")]="{ right: .7em; left:auto}",o[e("MENUINPUTBOX")]="{ padding-left: 1em; right:.5em; color:#666666; font-family: null;}",o[e("MENUINPUTBOX")+e("RTL")]="{ left: .1em;}",o[e("MENUCOMBOBOX")]="{ left:.1em; padding-bottom:.5em;}",o[e("MENUSLIDER")]="{ left: .1em;}",o[e("SLIDERVALUE")]="{ position:absolute; right:.1em; padding-top:.25em; color:#333333; font-size: .75em}",o[e("SLIDERBAR")]="{ outline: none; background: #d3d3d3}",o[e("MENULABEL")]="{ padding: 1px 2em 3px 1.33em; font-style:italic}",o[e("MENURULE")]="{ border-top: 1px solid #DDDDDD; margin: 4px 3px;}",o[e("MENUDISABLED")]="{ color:GrayText}",o[e("MENUACTIVE")]="{ background-color: #606872; color: white;}",o[e("MENUDISABLED")+":focus"]="{ background-color: #E8E8E8}",o[e("MENULABEL")+":focus"]="{ background-color: #E8E8E8}",o[e("CONTEXTMENU")+":focus"]="{ outline:none}",o[e("CONTEXTMENU")+" "+e("MENUITEM")+":focus"]="{ outline:none}",o[e("SELECTIONMENU")]="{ position:relative; float:left; border-bottom: none; -webkit-box-shadow:none; -webkit-border-radius:0px; }",o[e("SELECTIONITEM")]="{ padding-right: 1em;}",o[e("SELECTION")]="{ right: 40%; width:50%; }",o[e("SELECTIONBOX")]="{ padding: 0em; max-height:20em; max-width: none; background-color:#FFFFFF;}",o[e("SELECTIONDIVIDER")]="{ clear: both; border-top: 2px solid #000000;}",o[e("MENU")+" "+e("MENUCLOSE")]="{ top:-10px; left:-10px}";var i={};i[e("MENUCLOSE")]='{ position:absolute; cursor:pointer; display:inline-block; border:2px solid #AAA; border-radius:18px; -webkit-border-radius: 18px; /* Safari and Chrome */ -moz-border-radius: 18px; /* Firefox */ -khtml-border-radius: 18px; /* Konqueror */ font-family: "Courier New", Courier; font-size:24px; color:#F0F0F0}',i[e("MENUCLOSE")+" span"]="{ display:block; background-color:#AAA; border:1.5px solid; border-radius:18px; -webkit-border-radius: 18px; /* Safari and Chrome */ -moz-border-radius: 18px; /* Firefox */ -khtml-border-radius: 18px; /* Konqueror */ line-height:0; padding:8px 0 6px /* may need to be browser-specific */}",i[e("MENUCLOSE")+":hover"]="{ color:white!important; border:2px solid #CCC!important}",i[e("MENUCLOSE")+":hover span"]="{ background-color:#CCC!important}",i[e("MENUCLOSE")+":hover:focus"]="{ outline:none}";var a=!1,s=!1,l=!1;function c(t){l||(u(i,t),l=!0)}function u(t,e){var r=e||document,n=r.createElement("style");n.type="text/css";var o="";for(var i in t)o+=i,o+=" ",o+=t[i],o+="\n";n.innerHTML=o,r.head.appendChild(n)}t.addMenuStyles=function(t){s||(u(o,t),s=!0,c(t))},t.addInfoStyles=function(t){a||(u(r,t),a=!0,c(t))}}(e.CssStyles||(e.CssStyles={}))},2165:function(t,e){Object.defineProperty(e,"__esModule",{value:!0}),e.HtmlAttrs=e.HtmlClasses=void 0;function r(t){return"CtxtMenu_"+t}function n(t){return r(t)}function o(t){return r(t)}e.HtmlClasses={ATTACHED:n("Attached"),CONTEXTMENU:n("ContextMenu"),MENU:n("Menu"),MENUARROW:n("MenuArrow"),MENUACTIVE:n("MenuActive"),MENUCHECK:n("MenuCheck"),MENUCLOSE:n("MenuClose"),MENUCOMBOBOX:n("MenuComboBox"),MENUDISABLED:n("MenuDisabled"),MENUFRAME:n("MenuFrame"),MENUITEM:n("MenuItem"),MENULABEL:n("MenuLabel"),MENURADIOCHECK:n("MenuRadioCheck"),MENUINPUTBOX:n("MenuInputBox"),MENURULE:n("MenuRule"),MENUSLIDER:n("MenuSlider"),MOUSEPOST:n("MousePost"),RTL:n("RTL"),INFO:n("Info"),INFOCLOSE:n("InfoClose"),INFOCONTENT:n("InfoContent"),INFOSIGNATURE:n("InfoSignature"),INFOTITLE:n("InfoTitle"),SLIDERVALUE:n("SliderValue"),SLIDERBAR:n("SliderBar"),SELECTION:n("Selection"),SELECTIONBOX:n("SelectionBox"),SELECTIONMENU:n("SelectionMenu"),SELECTIONDIVIDER:n("SelectionDivider"),SELECTIONITEM:n("SelectionItem")},e.HtmlAttrs={COUNTER:o("Counter"),KEYDOWNFUNC:o("keydownFunc"),CONTEXTMENUFUNC:o("contextmenuFunc"),OLDTAB:o("Oldtabindex"),TOUCHFUNC:o("TouchFunc")}},4922:function(t,e,r){var n,o=this&&this.__extends||(n=function(t,e){return(n=Object.setPrototypeOf||{__proto__:[]}instanceof Array&&function(t,e){t.__proto__=e}||function(t,e){for(var r in e)e.hasOwnProperty(r)&&(t[r]=e[r])})(t,e)},function(t,e){function r(){this.constructor=t}n(t,e),t.prototype=null===e?Object.create(e):(r.prototype=e.prototype,new r)});Object.defineProperty(e,"__esModule",{value:!0}),e.Info=void 0;var i=r(5179),a=r(2165),s=function(t){function e(e,r,n){var o=t.call(this)||this;return o.title=e,o.signature=n,o.className=a.HtmlClasses.INFO,o.role="dialog",o.contentDiv=o.generateContent(),o.close=o.generateClose(),o.content=r||function(){return""},o}return o(e,t),e.prototype.attachMenu=function(t){this.menu=t},e.prototype.generateHtml=function(){t.prototype.generateHtml.call(this);var e=this.html;e.appendChild(this.generateTitle()),e.appendChild(this.contentDiv),e.appendChild(this.generateSignature()),e.appendChild(this.close.html),e.setAttribute("tabindex","0")},e.prototype.post=function(){t.prototype.post.call(this);var e=document.documentElement,r=this.html,n=window.innerHeight||e.clientHeight||e.scrollHeight||0,o=Math.floor(-r.offsetWidth/2),i=Math.floor((n-r.offsetHeight)/3);r.setAttribute("style","margin-left: "+o+"px; top: "+i+"px;"),window.event instanceof MouseEvent&&r.classList.add(a.HtmlClasses.MOUSEPOST),r.focus()},e.prototype.display=function(){this.menu.registerWidget(this),this.contentDiv.innerHTML=this.content();var t=this.menu.html;t.parentNode&&t.parentNode.removeChild(t),this.menu.frame.appendChild(this.html)},e.prototype.click=function(t){},e.prototype.keydown=function(e){this.bubbleKey(),t.prototype.keydown.call(this,e)},e.prototype.escape=function(t){this.unpost()},e.prototype.unpost=function(){t.prototype.unpost.call(this),this.html.classList.remove(a.HtmlClasses.MOUSEPOST),this.menu.unregisterWidget(this)},e.prototype.generateClose=function(){var t=new i.CloseButton(this),e=t.html;return e.classList.add(a.HtmlClasses.INFOCLOSE),e.setAttribute("aria-label","Close Dialog Box"),t},e.prototype.generateTitle=function(){var t=document.createElement("span");return t.innerHTML=this.title,t.classList.add(a.HtmlClasses.INFOTITLE),t},e.prototype.generateContent=function(){var t=document.createElement("div");return t.classList.add(a.HtmlClasses.INFOCONTENT),t.setAttribute("tabindex","0"),t},e.prototype.generateSignature=function(){var t=document.createElement("span");return t.innerHTML=this.signature,t.classList.add(a.HtmlClasses.INFOSIGNATURE),t},e.prototype.toJson=function(){return{type:""}},e}(r(8372).AbstractPostable);e.Info=s},1409:function(t,e,r){var n,o=this&&this.__extends||(n=function(t,e){return(n=Object.setPrototypeOf||{__proto__:[]}instanceof Array&&function(t,e){t.__proto__=e}||function(t,e){for(var r in e)e.hasOwnProperty(r)&&(t[r]=e[r])})(t,e)},function(t,e){function r(){this.constructor=t}n(t,e),t.prototype=null===e?Object.create(e):(r.prototype=e.prototype,new r)});Object.defineProperty(e,"__esModule",{value:!0}),e.Checkbox=void 0;var i=r(6765),a=r(2556),s=r(2165),l=function(t){function e(e,r,n,o){var i=t.call(this,e,"checkbox",r,o)||this;return i.role="menuitemcheckbox",i.variable=e.pool.lookup(n),i.register(),i}return o(e,t),e.fromJson=function(t,e,r){return new this(r,e.content,e.variable,e.id)},e.prototype.executeAction=function(){this.variable.setValue(!this.variable.getValue()),a.MenuUtil.close(this)},e.prototype.generateSpan=function(){this.span=document.createElement("span"),this.span.textContent="\u2713",this.span.classList.add(s.HtmlClasses.MENUCHECK)},e.prototype.updateAria=function(){this.html.setAttribute("aria-checked",this.variable.getValue()?"true":"false")},e.prototype.updateSpan=function(){this.span.style.display=this.variable.getValue()?"":"none"},e.prototype.toJson=function(){return{type:""}},e}(i.AbstractVariableItem);e.Checkbox=l},9886:function(t,e,r){var n,o=this&&this.__extends||(n=function(t,e){return(n=Object.setPrototypeOf||{__proto__:[]}instanceof Array&&function(t,e){t.__proto__=e}||function(t,e){for(var r in e)e.hasOwnProperty(r)&&(t[r]=e[r])})(t,e)},function(t,e){function r(){this.constructor=t}n(t,e),t.prototype=null===e?Object.create(e):(r.prototype=e.prototype,new r)});Object.defineProperty(e,"__esModule",{value:!0}),e.Combo=void 0;var i=r(6765),a=r(2556),s=r(2165),l=r(3205),c=function(t){function e(e,r,n,o){var i=t.call(this,e,"combobox",r,o)||this;return i.role="combobox",i.inputEvent=!1,i.variable=e.pool.lookup(n),i.register(),i}return o(e,t),e.fromJson=function(t,e,r){return new this(r,e.content,e.variable,e.id)},e.prototype.executeAction=function(){this.variable.setValue(this.input.value,a.MenuUtil.getActiveElement(this))},e.prototype.space=function(e){t.prototype.space.call(this,e),a.MenuUtil.close(this)},e.prototype.focus=function(){t.prototype.focus.call(this),this.input.focus()},e.prototype.unfocus=function(){t.prototype.unfocus.call(this),this.updateSpan()},e.prototype.generateHtml=function(){t.prototype.generateHtml.call(this),this.html.classList.add(s.HtmlClasses.MENUCOMBOBOX)},e.prototype.generateSpan=function(){this.span=document.createElement("span"),this.span.classList.add(s.HtmlClasses.MENUINPUTBOX),this.input=document.createElement("input"),this.input.addEventListener("keydown",this.inputKey.bind(this)),this.input.setAttribute("size","10em"),this.input.setAttribute("type","text"),this.input.setAttribute("tabindex","-1"),this.span.appendChild(this.input)},e.prototype.inputKey=function(t){this.bubbleKey(),this.inputEvent=!0},e.prototype.keydown=function(e){if(this.inputEvent&&e.keyCode!==l.KEY.ESCAPE&&e.keyCode!==l.KEY.RETURN)return this.inputEvent=!1,void e.stopPropagation();t.prototype.keydown.call(this,e),e.stopPropagation()},e.prototype.updateAria=function(){},e.prototype.updateSpan=function(){var t;try{t=this.variable.getValue(a.MenuUtil.getActiveElement(this))}catch(e){t=""}this.input.value=t},e.prototype.toJson=function(){return{type:""}},e}(i.AbstractVariableItem);e.Combo=c},3467:function(t,e,r){var n,o=this&&this.__extends||(n=function(t,e){return(n=Object.setPrototypeOf||{__proto__:[]}instanceof Array&&function(t,e){t.__proto__=e}||function(t,e){for(var r in e)e.hasOwnProperty(r)&&(t[r]=e[r])})(t,e)},function(t,e){function r(){this.constructor=t}n(t,e),t.prototype=null===e?Object.create(e):(r.prototype=e.prototype,new r)});Object.defineProperty(e,"__esModule",{value:!0}),e.Command=void 0;var i=r(1340),a=r(2556),s=function(t){function e(e,r,n,o){var i=t.call(this,e,"command",r,o)||this;return i.command=n,i}return o(e,t),e.fromJson=function(t,e,r){return new this(r,e.content,e.action,e.id)},e.prototype.executeAction=function(){try{this.command(a.MenuUtil.getActiveElement(this))}catch(t){a.MenuUtil.error(t,"Illegal command callback.")}a.MenuUtil.close(this)},e.prototype.toJson=function(){return{type:""}},e}(i.AbstractItem);e.Command=s},2965:function(t,e,r){var n,o=this&&this.__extends||(n=function(t,e){return(n=Object.setPrototypeOf||{__proto__:[]}instanceof Array&&function(t,e){t.__proto__=e}||function(t,e){for(var r in e)e.hasOwnProperty(r)&&(t[r]=e[r])})(t,e)},function(t,e){function r(){this.constructor=t}n(t,e),t.prototype=null===e?Object.create(e):(r.prototype=e.prototype,new r)});Object.defineProperty(e,"__esModule",{value:!0}),e.Label=void 0;var i=r(1340),a=r(2165),s=function(t){function e(e,r,n){return t.call(this,e,"label",r,n)||this}return o(e,t),e.fromJson=function(t,e,r){return new this(r,e.content,e.id)},e.prototype.generateHtml=function(){t.prototype.generateHtml.call(this),this.html.classList.add(a.HtmlClasses.MENULABEL)},e.prototype.toJson=function(){return{type:""}},e}(i.AbstractItem);e.Label=s},385:function(t,e,r){var n,o=this&&this.__extends||(n=function(t,e){return(n=Object.setPrototypeOf||{__proto__:[]}instanceof Array&&function(t,e){t.__proto__=e}||function(t,e){for(var r in e)e.hasOwnProperty(r)&&(t[r]=e[r])})(t,e)},function(t,e){function r(){this.constructor=t}n(t,e),t.prototype=null===e?Object.create(e):(r.prototype=e.prototype,new r)});Object.defineProperty(e,"__esModule",{value:!0}),e.Radio=void 0;var i=r(6765),a=r(2556),s=r(2165),l=function(t){function e(e,r,n,o){var i=t.call(this,e,"radio",r,o)||this;return i.role="menuitemradio",i.variable=e.pool.lookup(n),i.register(),i}return o(e,t),e.fromJson=function(t,e,r){return new this(r,e.content,e.variable,e.id)},e.prototype.executeAction=function(){this.variable.setValue(this.id),a.MenuUtil.close(this)},e.prototype.generateSpan=function(){this.span=document.createElement("span"),this.span.textContent="\u2713",this.span.classList.add(s.HtmlClasses.MENURADIOCHECK)},e.prototype.updateAria=function(){this.html.setAttribute("aria-checked",this.variable.getValue()===this.id?"true":"false")},e.prototype.updateSpan=function(){this.span.style.display=this.variable.getValue()===this.id?"":"none"},e.prototype.toJson=function(){return{type:""}},e}(i.AbstractVariableItem);e.Radio=l},3463:function(t,e,r){var n,o=this&&this.__extends||(n=function(t,e){return(n=Object.setPrototypeOf||{__proto__:[]}instanceof Array&&function(t,e){t.__proto__=e}||function(t,e){for(var r in e)e.hasOwnProperty(r)&&(t[r]=e[r])})(t,e)},function(t,e){function r(){this.constructor=t}n(t,e),t.prototype=null===e?Object.create(e):(r.prototype=e.prototype,new 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i=r(6765),a=r(2556),s=r(2165),l=r(3205),c=function(t){function e(e,r,n,o){var i=t.call(this,e,"slider",r,o)||this;return i.role="slider",i.labelId="ctx_slideLabel"+a.MenuUtil.counter(),i.valueId="ctx_slideValue"+a.MenuUtil.counter(),i.inputEvent=!1,i.variable=e.pool.lookup(n),i.register(),i}return o(e,t),e.fromJson=function(t,e,r){return new 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