
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
Triangle  $ABC$  satisfies  $AB=104$ ,  $BC=112$ , and  $CA=120$ .  Let  $\omega$  and  $\omega_A$  denote the incircle and  $A$ -excircle of  $\triangle ABC$ , respectively.  There exists a unique circle  $\Omega$  passing through  $A$  which is internally tangent to  $\omega$  and externally tangent to  $\omega_A$ .  Compute the radius of  $\Omega$ .

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