Positional Grouping Problem - Add hierarchy to a flat structure

I currently have an xml file (a series of sequential statements printf'ed)
that looks like this (simplified example):

<LogFile>
    <BeginFile lineNum="1" fileName="file1.txt"/>
    <ElementA lineNum="2"/>
    <ElementB lineNum="3"/>
    <ElementC lineNum="4"/>
    <BeginFile lineNum="6" fileName="file2.txt"/>
    <ElementA lineNum="7"/>
    <ElementB lineNum="8"/>
    <BeginFile lineNum="9" fileName="file2b.txt"/>
    <ElementA lineNum="10"/>
    <ElementB lineNum="11"/>
    <EndFile lineNum="12" fileName="file2b.txt"/>
    <ElementA lineNum="13"/>
    <ElementB lineNum="14"/>
    <ElementC lineNum="15"/>
    <EndFile lineNum="16" fileName="file1.txt"/>
</LogFile>

And would like to transform it to look like this:
<LogFile>
    <File lineNum="1" fileName="file1.txt"/>
        <ElementA lineNum="2"/>
        <ElementB lineNum="3"/>
        <ElementC lineNum="4"/>
        <File lineNum="6" fileName="file2.txt"/>
            <ElementA lineNum="7"/>
            <ElementB lineNum="8"/>
            <File lineNum="9" fileName="file2b.txt"/>
                <ElementA lineNum="10"/>
                <ElementB lineNum="11"/>
            </File>
        <ElementA lineNum="13"/>
        <ElementB lineNum="14"/>
        <ElementC lineNum="15"/>
    </File>
</LogFile>

Basically, I want to replace the <BeginFile> node, with a <File> node.
And then copy all elements between <BeginFile file="x"> and its
corresponding <EndFile file="x"> as subelements to this new <File> node.
Nesting <File> nodes if a second <BeginFile> node is found before the
corresponding <EndFile> node is found. For each <BeginFile file="x"/>, there
will always be a corresponding <EndFile file="x"/> element. And the LineNum
attribute is not that important, but are in the original document, are
valid, and can be used in the solution.

I think the solution should be simple, but I am having a brain freeze.
It is similar to this:
        http://www.dpawson.co.uk/xsl/sect2/N4486.html#d4110e229
But the hierarchy <File> nodes and the intermediate <ElementXX> nodes are
confusing me.

Any help is appreciated.
Tx in Advance.
Tom

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