
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 the cube $ABCD-A_{1}B_{1}C_{1}D_{1}$ with edge length $10$, $P$ is a point inside the side face $ADD_{1}A_{1}$. The distances from $P$ to $A_{1}D_{1}$ and $AA_{1}$ are $3$ and $2$ respectively. A line passing through point $P$ and parallel to $A_{1}C$ intersects the surface of the cube at another point $Q$. Find the value of $|PQ|$.

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