
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
Given rectangle ABCD with AB = 10 and BC = 8, points E and F lie on AB, points G and H lie on BC, points I and J lie on CD, and points K and L lie on DA such that AE = FB < 5 and the convex octagon EFGHIJKL is equilateral. Determine the length of a side of this octagon in the form $p+q\sqrt{r}$, where p, q, and r are integers and r is not divisible by the square of any prime.

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