
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
Given the vertex of a cone, \( P \), an isosceles right triangle with the axis section, point \( A \) on the circumference of the cone's base, point \( B \) inside the base circle, and center of the base circle \( O \), where \( AB \perp OB \) 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 \( OB \) when the volume of the tetrahedron \( OHPC \) is maximized.

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