
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
The $7\times21$ rectangle $ABCD$ is cut into two congruent hexagons in such a way that the two hexagons can be repositioned without overlap to form a square. Additionally, it's given that one of the dimensions of each hexagon, $z$, is equal to one-third the side of the resulting square. What is $z$?
A) $\frac{5\sqrt{3}}{3}$
B) $\frac{7\sqrt{3}}{3}$
C) $\frac{7\sqrt{3}}{2}$
D) $\frac{9\sqrt{3}}{3}$
E) $\frac{7\sqrt{5}}{3}$

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