
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 $\triangle ABC$ be an equilateral triangle where each side has length $a$. Let $P$ be a point on the side $AB$ such that $AP = x$ and $PB = a - x$. Calculate the expression $t = AP^2 + PB^2 + CP^2$ and determine when $t < a^2$, $t = a^2$, $t > a^2$, $t = 1.5a^2$.

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