
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 cross-section of a cone with vertex \( P \) is an isosceles right triangle. \( A \) is a point on the circular base's circumference, \( B \) is a point within the circular base, \( O \) is the center of the circular base, \( AB \perp OB \) with \( B \) as the foot of the perpendicular. \( OH \perp PB \) with \( H \) as the foot of this perpendicular, and \( PA = 4 \). \( C \) is the midpoint of \( PA \). Find the length of \( OB \) such that the volume of the tetrahedron \( O-HPC \) is maximized.

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