
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 semicircle with diameter $AB$ of 10 cm. A square $ABCD$ is constructed outside the semicircle such that $AB$ is the diameter of the semicircle. If $N$ is the point on the semicircle such that $AN = NB$, locate the midpoint $M$ of $AB$, and determine the length of segment $NC$. Provide your answer in simplest radical form.

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