
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
In rectangle $ABCD$, $AB = 10$ and $BC = 12$. Points $E$ and $F$ lie on $\overline{AB}$, points $G$ and $H$ lie on $\overline{BC}$, points $I$ and $J$ lie on $\overline{CD}$, and points $K$ and $L$ lie on $\overline{DA}$ such that $AE = FB < 5$ and the convex octagon $EFGHIJKL$ has all sides equal except $EF$ and $KL$ which are shorter. Calculate $EF$ when the length of the other six sides of the octagon can be expressed 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.
A) $10.5 - \sqrt{121}$
B) $10 - \sqrt{244}$
C) $10.5 - \frac{\sqrt{242}}{2}$
D) $11 - \sqrt{250}$

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