
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
Let  $ABCD$  be a rectangle and  $E$  be a point on segment  $AD$ . We are given that quadrilateral  $BCDE$  has an inscribed circle  $\omega_1$  that is tangent to  $BE$  at  $T$ . If the incircle  $\omega_2$  of  $ABE$  is also tangent to  $BE$  at  $T$ , then find the ratio of the radius of  $\omega_1$  to the radius of  $\omega_2$ .

*Proposed by James Lin.*

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