
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
In  $\triangle ABC$  with  $AB = 10$ ,  $BC = 12$ , and  $AC = 14$ , let  $E$  and  $F$  be the midpoints of  $AB$  and  $AC$ . If a circle passing through  $B$  and  $C$  is tangent to the circumcircle of  $AEF$  at point  $X \ne A$ , find  $AX$ .

*Proposed by Vivian Loh*

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