
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
In a cube \( A B C D - A_{1} B_{1} C_{1} D_{1} \), let the points \( P \) and \( Q \) be the reflections of vertex \( A \) with respect to the plane \( C_{1} B D \) and the line \( B_{1} D \), respectively. Find the sine of the angle between the line \( P Q \) and the plane \( A_{1} B D \).

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