
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 axial section of a cone with vertex \( P \) is an isosceles right triangle. Point \( A \) is on the circumference of the base, point \( B \) is inside the base circle, point \( O \) is the center of the base circle, and \( AB \perp OB \) with the foot of the perpendicular at \( B \). \( OH \perp PB \) with the foot of the perpendicular at \( H \), and \( PA = 4 \). Point \( C \) is the midpoint of \( PA \). Determine 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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