
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
A circle is centered at point $O$ with diameter $AB = 8$ cm also serving as one side of square $ABCD$. If $N$ is the midpoint of diagonal $AC$, and $P$ is the midpoint of arc $AB$ that does not contain $O$, what is the length of segment $PN$?

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