
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 $P$ be a point on side $AC$ of an equilateral triangle $ABC$ with side length $a$. Define $s = AP^2 + PC^2$ and analyze $t = BP^2$.
A) $s > t$ for all positions of $P$ on $AC$
B) $s = t$ only if $P$ coincides with $A$ or $C$
C) $s < t$ for all positions of $P$ on $AC$
D) $s < t$ if $0 < x < a$
E) $s = t$ if $0 < x < a$

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