Hello I am trying to validate account name. I want to check if the account with same name exists so i have written controller class method to check it but i am not able to call it from visuaforce page.
Following is my visualforce page

<apex:page controller="AddAccountController" tabStyle="Account" standardStylesheets="false" showHeader="false" sidebar="false">
    <style>
        .myFormStyle {
        background-color: #D3D3D3;
        }
        .tableBorder
        {
        
        border:3px outset black;
        }
        .innerTableBorder{
        border-top:2px dotted black;
        border-left:2px dotted black;
        }
    </style>
    <script>
     function AccountCheck() {
         
         var x = "{!checkAccount}"
       alert(x);
        }
    </script>
    
    
    
    <apex:form styleClass="myFormStyle">
        <div>
            <apex:pageBlock title="Add Account">
                <apex:pageBlockButtons location="top" >                   
                    <apex:commandButton value="Save" action="{!save}" reRender="accountList"  />
                    <apex:commandButton value="Cancel" action="{!cancel}" />
                    
                </apex:pageBlockButtons>  
                <apex:pageBlockSection title="Account Details" columns="1">            
                    <apex:inputField id="accountName" value="{!act.name}" onblur="AccountCheck()"/> 
                    <apex:inputField value="{!act.site}"/> 
                    <apex:inputField value="{!act.type}"/> 
                    <apex:inputField value="{!act.accountNumber}"/>              
                </apex:pageBlockSection>         
            <apex:pageBlockTable value="{!Account}" var="acc" id="accountList" styleClass="tableBorder">              
                <apex:column value="{!acc.Name}" styleClass="innerTableBorder" />            
                <apex:column value="{!acc.Site}" styleClass="innerTableBorder" />               
                <apex:column value="{!acc.type}" styleClass="innerTableBorder" />               
            </apex:pageBlockTable>
        </apex:pageBlock>
    </div>
</apex:form>
</apex:page>

Here on its my Controller
 

public class AddAccountController {
    
    public List <Account> account=new List <Account>();
    public Account act{get;set;}
    public integer counter {get;set;}
    
    public AddAccountController() {       
        act = new Account();
        this.counter=0;
    }
    
    public List <Account> getAccount() {
        account = [SELECT Id, Name, Site, Type FROM Account order by createddate desc limit 20];
        return account;
    }
    
    public PageReference cancel() {
        return null;
    }
    
    public PageReference save() {
        system.debug('act name:' + act.name + ' = ' + act.accountNumber);
        upsert act;
        return null;
    }
    public Integer checkAccount(){
        String accountName =System.currentPageReference().getParameters().get('accountName');
        account = [SELECT Id, Name, Site, Type FROM Account WHERE Name = :accountName];
        if(account.size()>0){
            counter=1;
            return counter;
        }
        return counter;
    }
}

Below is the error that i get please tell me why i get this error.

Unknown property 'AddAccountController.checkAccount'

I want it to call the checkAccount() method  from my javasccript function that i am calling from onblur method. So please experts help a newbie