
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
Square  $ABCD$  has side length  $1$ ; circle  $\Gamma$  is centered at  $A$  with radius  $1$ . Let  $M$  be the midpoint of  $BC$ , and let  $N$  be the point on segment  $CD$  such that  $MN$  is tangent to  $\Gamma$ . Compute  $MN$ .

*2018 CCA Math Bonanza Individual Round #11*

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