
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
Let  $A,B,C$  be nodes of the lattice  $Z\times Z$  such that inside the triangle  $ABC$  lies a unique node  $P$  of the lattice. Denote  $E = AP \cap BC$ . Determine max  $\frac{AP}{PE}$  , over all such configurations.

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