
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
Inside the triangle \(ABC\), there are points \(P\) and \(Q\) such that point \(P\) is at distances 6, 7, and 12 from lines \(AB\), \(BC\), and \(CA\) respectively, and point \(Q\) is at distances 10, 9, and 4 from lines \(AB\), \(BC\), and \(CA\) respectively. Find the radius of the inscribed circle of triangle \(ABC\).

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