
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 round table has a radius of $5$. Eight rectangular placemats are placed on the table. Each placemat has a width of $1$ and a length $y$. They are positioned so that each mat has two corners on the edge of the table, and these corners are endpoints of the same side of length $y$. Additionally, the mats are arranged so that the inner corners of each mat touch the inner corners of adjacent mats. Determine the value of $y$.

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