
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 $I$ be the incenter of triangle $ABC,$ and $P$ be an arbitrary point. Determine a constant $k$ such that
\[PA^2 + PB^2 + PC^2 = k \cdot PI^2 + IA^2 + IB^2 + IC^2.\]

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