
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  $ABC$  be a triangle with sides  $AB=19$ ,  $BC=21$ , and  $AC=20$ . Let  $\omega$  be the incircle of  $ABC$  with center  $I$ . Extend  $BI$  so that it intersects  $AC$  at  $E$ . If  $\omega$  is tangent to  $AC$  at the point  $D$ , then compute the length of  $DE$ .

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