
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  $ABC$ ,  $AB=3$ ,  $AC=5$ , and  $BC=7$ . Let  $E$  be the reflection of  $A$  over  $\overline{BC}$ , and let line  $BE$  meet the circumcircle of  $ABC$  again at  $D$ . Let  $I$  be the incenter of  $\triangle ABD$ . Given that  $\cos ^2  \angle AEI = \frac{m}{n},$  where  $m$  and  $n$  are relatively prime positive integers, determine  $m+n$ .

*Proposed by Ray Li*

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