
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
Triangle  $ABC$  with  $\measuredangle A = 90^\circ$  has incenter  $I$ . A circle passing through  $A$  with center  $I$  is drawn, intersecting  $\overline{BC}$  at  $E$  and  $F$  such that  $BE < BF$ . If  $\tfrac{BE}{EF} = \tfrac{2}{3}$ , then  $\tfrac{CF}{FE} = \tfrac{m}{n}$ , where  $m$  and  $n$  are relatively prime positive integers. Find  $m+n$ .

*Proposed by Lewis Chen*

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