A4J:support doesn't work when javascript event is setted into h:inputText

Question

I'm maintaining a JSF legacy system and I have a following problem: There is a:inputText component with events onchange, onkeyup and onkeydown triggering javascript function, like this:

<h:inputText id="idX" 
    value="#{myBackingbean.myProperty}"
    maxlength="14" size="14"
    onkeypress="return javascriptMethodOne(event);"
    onkeyup="javascriptMethodTwo(this, 1), javascriptMethodThree('idX','other_ID',14,event);"
    onchange="javascriptMethodTwo(this, 1); javascriptMethodFour(JS_CONST);" />

I need call a backing bean method from h:inputText and I thought add a a4j:support into h:inputText, like this code:

<h:inputText ... >
    <a4j:support .../>
</h:inputText>

The problem is a4j:support doesn't work when events are setted in componet h:inputText.

So, keeping the calls of javascript functions, how I can do to call a backing bean method when onchange, onkeyup or onkeydown is triggered?

The following code doesn't work:

<h:inputText id="idX" 
    value="#{myBackingbean.myProperty}"
    maxlength="14" size="14"
    onkeypress="return javascriptMethodOne(event);"
    onkeyup="javascriptMethodTwo(this, 1), javascriptMethodThree('idX','other_ID',14,event);"
    onchange="javascriptMethodTwo(this, 1); javascriptMethodFour(JS_CONST);">

<a4j:support action="#{myBackingbean.myMethod}" event="onchange"/>

    <a4j:support action="#{myBackingbean.myMethod}" event="onkeyup"/>

    <a4j:support action="#{myBackingbean.myMethod}" event="onkeydown"/>

</h:inputText>

Thx!