
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 a point \( P \) on the surface of a sphere with radius 5, three mutually perpendicular chords \( P A, P B, \) and \( P C \) are drawn such that \( P A = 2 P B \). What is the maximum value of \( P A + P B + P C \)?

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