
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  $ABC$  be a triangle with circumcenter  $O$  and incenter  $I$ , and suppose that  $OI$  meets  $AB$  and  $AC$  at  $P$  and  $Q$ , respectively. There exists a point  $R$  on arc  $\widehat{BAC}$  such that the circumcircles of triangles  $PQR$  and  $ABC$  are tangent. Given that  $AB = 14$ ,  $BC = 20$ , and  $CA = 26$ , find  $\frac{RC}{RB}$ .

*Proposed by Andrew Wu*

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