
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
Let  $ABCD$  be a quadrilateral with an inscribed circle  $\omega$  and let  $P$  be the intersection of its diagonals  $AC$  and  $BD$ . Let  $R_1$ ,  $R_2$ ,  $R_3$ ,  $R_4$  be the circumradii of triangles  $APB$ ,  $BPC$ ,  $CPD$ ,  $DPA$  respectively. If  $R_1=31$  and  $R_2=24$  and  $R_3=12$ , find  $R_4$ .

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