
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
Rectangle $ABCD$ has an area of $32$, and side $\overline{AB}$ is parallel to the x-axis. Side $AB$ measures $8$ units. Vertices $A,$ $B$, and $C$ are located on the graphs of $y = \log_a x$, $y = 2\log_a x$, and $y = 4\log_a x$, respectively. Determine the value of $a$.
A) $\sqrt[3]{\frac{1 + \sqrt{33}}{2} + 8}$
B) $\sqrt[4]{\frac{1 + \sqrt{33}}{2} + 8}$
C) $\sqrt{\frac{1 + \sqrt{33}}{2} + 8}$
D) $\sqrt[6]{\frac{1 + \sqrt{33}}{2} + 8}$
E) $\sqrt[5]{\frac{1 + \sqrt{43}}{2} + 8}$

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