
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  $ABC$ ,  $AB = 100$ ,  $BC = 120$ , and  $CA = 140$ . Points  $D$  and  $F$  lie on  $\overline{BC}$  and  $\overline{AB}$ , respectively, such that  $BD = 90$  and  $AF = 60$ . Point  $E$  is an arbitrary point on  $\overline{AC}$ . Denote the intersection of  $\overline{BE}$  and  $\overline{CF}$  as  $K$ , the intersection of  $\overline{AD}$  and  $\overline{CF}$  as  $L$ , and the intersection of  $\overline{AD}$  and  $\overline{BE}$  as  $M$ . If  $[KLM] = [AME] + [BKF] + [CLD]$ , where  $[X]$  denotes the area of region  $X$ , compute  $CE$ .

*Proposed by Lewis Chen*

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