
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 $G$ be the centroid of triangle $ABC$. For all points $P$ on the circumcircle of triangle $ABC$,
\[PA^2 + PB^2 + PC^2 - PG^2\]
is a constant. Express this constant in terms of the side lengths $a$, $b$, $c$, and the circumradius $R$ of triangle $ABC$.

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