
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 two squares $ABCD$ and $DCFE$ with side lengths of $1$, where the planes they reside in are perpendicular to each other. Points $P$ and $Q$ are moving points on line segments $BC$ and $DE$ (including endpoints), with $PQ = \sqrt{2}$. Let the trajectory of the midpoint of line segment $PQ$ be curve $\mathcal{A}$. Determine the length of $\mathcal{A}$.

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