
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 larger square contains two non-overlapping shapes: a circle with diameter $2$ and a rectangle with side lengths $2$ and $4$. Find the smallest possible side length of the larger square such that these shapes can fit without overlapping, and then, find the area of the square $S$ that can be inscribed precisely in the remaining free space inside the larger square.
A) $\sqrt{16-2\pi}$  
B) $\pi - \sqrt{8}$  
C) $\sqrt{8 - \pi}$  
D) $\sqrt{10 - \pi}$

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