
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 circular table has a radius of 6 units. Eight rectangular place mats are placed on it. Each mat has a width of 1 unit and a length of $y$ units, positioned so that two corners of the mat touch the table's edge, with these corners being endpoints of the side of length $y$. Additionally, the mats are arranged such that the inner corner of each touches an inner corner of adjacent mats. Compute $y$.
A) $6 \sqrt{2}$
B) $6 \sqrt{2 - \sqrt{2}}$
C) $6$
D) $6 \sqrt{2 + \sqrt{2}}$
E) $6 \sqrt{3}$

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