
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  $ABCD$  be a square with side length  $6$ . Circles  $X, Y$  , and  $Z$  are congruent circles with centers inside the square such that  $X$  is tangent to both sides  $\overline{AB}$  and  $\overline{AD}$ ,  $Y$  is tangent to both sides  $\overline{AB}$  and  $\overline{BC}$ , and  $Z$  is tangent to side  $\overline{CD}$  and both circles  $X$  and  $Y$  . The radius of the circle  $X$  can be written  $m -\sqrt{n}$ , where  $m$  and  $n$  are positive integers. Find  $m + n$ .

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