
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 square $ABCD$, points $M$ and $N$ are the midpoints of sides $\overline{BC}$ and $\overline{CD}$ respectively. Point $P$ is the midpoint of side $\overline{AB}$. Find $\cos \phi$ where $\phi$ is the angle $\angle APN$.

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