
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 axis section of the cone with vertex P is an isosceles right triangle. A is a point on the circumference of the base, B is a point inside the base circle, O is the center of the base circle, and AB is perpendicular to OB with the foot of the perpendicular being B. OH is perpendicular to PB with the foot of the perpendicular being H, and PA = 4. C is the midpoint of PA. Determine the length of OB when the volume of the tetrahedron C-HPC is maximized.

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