
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 $5$ units. Eight rectangular place mats are placed on the table. Each place mat has a width of $2$ units and a length $y$. They are positioned such that each mat has two corners on the edge of the table, and these two corners are endpoints of the same side of length $y$. Moreover, the mats are arranged such that their inner corners each touch an inner corner of two adjacent mats forming an octagon. Determine $y$.

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