
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 $BC=8$. Points $P$ and $Q$ lie on $\overline{AB}$, points $R$ and $S$ lie on $\overline{BC}$, points $T$ and $U$ lie on $\overline{CD}$, and points $V$ and $W$ lie on $\overline{DA}$ so that $AP=BQ<5$ and the convex octagon $PQRSTUVW$ is equilateral. The length of a side of this octagon can be expressed in the form $k+m\sqrt{n}$, where $k$, $m$, and $n$ are integers and $n$ is not divisible by the square of any prime. What is $k+m+n$?
A) 153
B) 154
C) 155
D) 156
E) 157

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