
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
Two circles \( C_1 \) and \( C_2 \) of radii \( 10 \) cm and \( 8 \) cm respectively are tangent to each other internally at a point \( A \). \( AD \) is the diameter of \( C_1 \) and \( P \) and \( M \) are points on \( C_1 \) and \( C_2 \) respectively such that \( PM \) is tangent to \( C_2 \). If \( PM = \sqrt{20} \) cm and \( \angle PAD = x^\circ \), find the value of \( x \).

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