
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Given a plane \(\alpha\) and a fixed point \(P\) outside \(\alpha\) with a distance \(h\) to \(\alpha\), three moving points \(A\), \(B\), and \(C\) on \(\alpha\) have distances to \(P\) denoted by \(a\), \(b\), and \(c\) respectively. If \(\angle PBA = 90^{\circ}\), then what is the maximum area of \(\triangle ABC\) in terms of \(a\), \(b\), \(c\), and \(h\)?

Present the answer in LaTex format: \boxed{Your answer}
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