
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
Given a square $ABCD$ with side length 12. Point $E$ is located on the extension of side $BC$ such that $BE = 16$. Point $P$ moves along $CE$ from $C$ to $E$. Construct square $APFG$ such that $A$ and $G$ are on the same side of $BE$. Determine the length of the trajectory of the center $H$ of square $APFG$ as $P$ moves from $C$ to $E$.

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