
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
Consider a  $1 \times 1$  grid of squares. Let  $A,B,C,D$  be the vertices of this square, and let  $E$  be the midpoint of segment  $CD$ . Furthermore, let  $F$   be the point on segment  $BC$  satisfying  $BF = 2CF$ , and let  $P$  be the intersection of lines  $AF$  and  $BE$ . Find  $\frac{AP}{PF}$ .

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