
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  $3 \times 3$  grid of squares. A circle is inscribed in the lower left corner, the middle square of the top row, and the rightmost square of the middle row, and a circle  $O$  with radius  $r$  is drawn such that  $O$  is externally tangent to each of the three inscribed circles. If the side length of each square is 1, compute  $r$ .

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