
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  $\omega_1$  be a circle of radius  $1$  that is internally tangent to a circle  $\omega_2$  of radius  $2$  at point  $A$ . Suppose  $\overline{AB}$  is a chord of  $\omega_2$  with length  $2\sqrt3$  that intersects  $\omega_1$  at point  $C\ne A$ . If the tangent line of  $\omega_1$  at  $C$  intersects  $\omega_2$  at points  $D$  and  $E$ , find  $CD^4 + CE^4$ .

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