
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
In \(\triangle ABC\), \(\tan \angle BCA = 1\) and \(\tan \angle BAC = \frac{1}{7}\). The perimeter of \(\triangle ABC\) is \(24 + 18 \sqrt{2}\). The altitude from \(B\) to \(AC\) has length \(h\) and intersects \(AC\) at \(D\). What is the value of \(h\)?

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