
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]


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Inside a square with side length 12, two congruent equilateral triangles are drawn such that they share one side and each has one vertex on a vertex of the square. Additionally, a circle is inscribed in the square such that it just touches each triangle at one point. What is the radius of the circle?

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