
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
The edge \( PA \) of the pyramid \( PABC \) is perpendicular to the base plane \( ABC \) and equals 1. In the triangle \( ABC \), the angle at vertex \( A \) is a right angle, and each leg \( AB \) and \( AC \) is equal to 2. Points \( M \) and \( N \) are the midpoints of \( AC \) and \( BC \) respectively. Find the radius of the sphere inscribed in the pyramid \( PMNC \).

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