
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 given configuration containing rectangle $ABCD$ with $AB = 6$ and $BC = 5$, segment $GH$ is constructed through $B$ such that $GH$ is perpendicular to $DB$, and $A$ and $C$ lie on $DG$ and $DH$ respectively. What is the length of $GH$?
A) $\frac{11\sqrt{55}}{6}$
B) $\frac{11\sqrt{61}}{6}$
C) $\frac{71}{6}$
D) $\frac{12\sqrt{61}}{7}$
E) $12$

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