
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
Circle  $\omega_1$  with radius 3 is inscribed in a strip  $S$  having border lines  $a$  and  $b$ .  Circle  $\omega_2$  within  $S$  with radius 2 is tangent externally to circle  $\omega_1$  and is also tangent to line  $a$ .  Circle  $\omega_3$  within  $S$  is tangent externally to both circles  $\omega_1$  and  $\omega_2$ , and is also tangent to line  $b$ .  Compute the radius of circle  $\omega_3$ .


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