
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
Three circles  $\Gamma_1,\Gamma_2,\Gamma_3$  are inscribed into an angle(the radius of  $\Gamma_1$  is the minimal, and the radius of  $\Gamma_3$  is the maximal) in such a way that  $\Gamma_2$  touches  $\Gamma_1$  and  $\Gamma_3$  at points  $A$  and  $B$  respectively. Let  $\ell$  be a tangent to  $A$  to  $\Gamma_1$ . Consider circles  $\omega$  touching  $\Gamma_1$  and  $\ell$ . Find the locus of meeting points of common internal tangents to  $\omega$  and  $\Gamma_3$ .

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