
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
Circles  $\omega_a, \omega_b, \omega_c$  have centers  $A, B, C$ , respectively and are pairwise externally tangent at points  $D, E, F$  (with  $D\in BC, E\in CA, F\in AB$ ). Lines  $BE$  and  $CF$  meet at  $T$ . Given that  $\omega_a$  has radius  $341$ , there exists a line  $\ell$  tangent to all three circles, and there exists a circle of radius  $49$  tangent to all three circles, compute the distance from  $T$  to  $\ell$ .

*Proposed by Andrew Gu.*

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