
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 a $30^\circ-60^\circ-90^\circ$ triangle with $AC$ being the hypotenuse. Let $AB = a$, $BC = a\sqrt{3}$, and $CA = 2a$. Place point $P$ on the line $AB$ or its extension, with $AP = x$ and $PB = a - x$ if $P$ is on $AB$, and $AP = a+x$, $PB = x$ if P is beyond $B$. Express $s = AP^2 + PB^2$ in terms of $CP^2$, where $CP^2$ is a function of $x$.

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