
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 a rectangle $WXYZ$, $WZ=10$ and $ZY=15$. Points $A$ and $B$ lie on $\overline{WZ}$, points $C$ and $D$ lie on $\overline{ZY}$, points $E$ and $F$ lie on $\overline{YX}$, and points $G$ and $H$ lie on $\overline{XW}$ so that $AW = BZ < 5$ and the convex octagon $ABCDEFGH$ 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) 33
B) 41
C) 50
D) 69

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