
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
\(ABCD\) is a square with a side length of \(20 \sqrt{5} x\). \(P\) and \(Q\) are the midpoints of \(DC\) and \(BC\) respectively.

1. If \(AP = ax\), find \(a\).
2. If \(PQ = b \sqrt{10} x\), find \(b\).
3. If the distance from \(A\) to \(PQ\) is \(c \sqrt{10} x\), find \(c\).
4. If \(\sin \theta = \frac{d}{100}\), find \(d\).

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