
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 square with a side length of 4. Four semicircles with radii 1 are inscribed inside the square, each fitting perfectly on one side of the square. Additionally, a quarter-circle with radius 2 is centered at one corner of the square. What is the radius of the smallest circle that is tangent to all four semicircles and the quarter-circle?
A) $\sqrt{2} - 1$
B) $\frac{3}{2}$
C) $\sqrt{2} - \frac{3}{2}$
D) 2
E) $\frac{5}{2} - \sqrt{2}$

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