
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
The cross-section along the axis of a cone with vertex \( P \) is an isosceles right triangle. Let \( A \) be a point on the circumference of the base, \( B \) be a point within the base circle such that \( AB \perp OB \) (where \( O \) is the center of the base circle) with the foot at \( B \), and \( OH \perp HB \) with the foot at \( H \). If \( PA = 4 \) and \( C \) is the midpoint of \( PA \), find the length of \( OB \) when the volume of the triangular pyramid \( O-HPC \) is maximized.

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