
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
Given is triangle  $ABC$  with incenter  $I$  and  $A$ -excenter  $J$ . Circle  $\omega_b$  centered at point  $O_b$  passes through point  $B$  and is tangent to line  $CI$  at point  $I$ . Circle  $\omega_c$  with center  $O_c$  passes through point  $C$  and touches line  $BI$  at point  $I$ . Let  $O_bO_c$  and  $IJ$  intersect at point  $K$ . Find the ratio  $IK/KJ$ .

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