
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
A rectangle has dimensions 4 (length) and 2 (height). Semicircles with equal radii line the inside of the lengthier side of the rectangle. Another circle inside the rectangle is tangent to all of these semicircles. Determine the radius of this inner tangent circle.
A) $\frac{\sqrt{6}-1}{2}$
B) $\frac{\sqrt{10}-1}{2}$
C) $\frac{\sqrt{8}-1}{2}$
D) $\frac{\sqrt{7}-1}{2}$
E) $\frac{\sqrt{9}-1}{2}$

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