
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  $\omega$  be a circle, and let  $ABCD$  be a quadrilateral inscribed in  $\omega$ .  Suppose that  $BD$  and  $AC$  intersect at a point  $E$ .  The tangent to  $\omega$  at  $B$  meets line  $AC$  at a point  $F$ , so that  $C$  lies between  $E$  and  $F$ .  Given that  $AE=6$ ,  $EC=4$ ,  $BE=2$ , and  $BF=12$ , find  $DA$ .

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