Country State City dropdown in jsf

Question

The city Dropdown populated when the state is selected and State Dropdown populated according Country selection; however, it is not doing this.

Problem is when country selected according country state populated in state dropdwon but after it when i select state from state dropdwon list it does not show populate city dropdwon

actual problem is State value not set in state variable when i select state

Here is the code:

JsfPage

 <p:selectOneMenu id="countrydrop" value="#{user.country}" >
            <f:ajax  id="countryajex" listener="#{user.statesbycontry()}" render="statedrop"/>
            <f:selectItem itemLabel="Select Country" itemValue="" noSelectionOption="true" />
            <f:selectItems value="#{user.allcontry()}" />
        </p:selectOneMenu>

        <p:selectOneMenu id="statedrop" value="#{user.ustate}" >
            <p:ajax event="change" id="stateajex" listener="#{user.citybystate()}" update="citydrop" />
            <f:selectItem itemLabel="Select State" itemValue="" noSelectionOption="true" />                
            <f:selectItems value="#{user.statelist}"/>                
        </p:selectOneMenu>

        <p:selectOneMenu id="citydrop" value="#{user.city}" >
            <f:selectItem itemLabel="Select City" itemValue="" noSelectionOption="true" />
            <f:selectItems value="#{user.citylist}" />
        </p:selectOneMenu> 

jsfManagedBean

String country; 
String state;       
String city;
List<String> statelist; 
List<String> citylist;


public String getCountry() {
    return country;
}

public void setCountry(String country) {
    this.country = country;
}

public String getState() {
    return state;
}

public void setState(String state) {
    this.state = state;
}

public String getCity() {
    return city;
}

public void setCity(String city) {
    this.city = city;
}

public List<String> getStatelist() {
    return statelist;
}

public void setStatelist(List<String> statelist) {
    this.statelist = statelist;
}

public List<String> getCitylist() {
    return citylist;
}

public void setCitylist(List<String> citylist) {
    this.citylist = citylist;
}  


public List<String> allcontry(){
return od_user_bean.allcountries();
} 


public void statesbycontry(){
//Method tested Working well
statelist = od_user_bean.findstatesbycontries(country);
} 


public void citybystate(){
    System.out.println("State="+state);
  if(state != null)
      //Method tested Working well
  citylist=od_user_bean.findcitybystate(state);
}