
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
Let \( O \) be the circumcenter, and \( H \) the orthocenter, of triangle \( ABC \) with side lengths \( a, b, c \) and circumradius \( R \). For any point \( P \) on the circumcircle of triangle \( ABC \), compute the quantity
\[
PA^2 + PB^2 + PC^2 + PH^2
\]
and express it in terms of \( a, b, c, \) and \( R \).

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