
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
Let $ABCD$ be an isosceles trapezoid with $\overline{AD}||\overline{BC}$ whose angle at the longer base $\overline{AD}$ is $\dfrac{\pi}{4}$. The diagonals have length $12\sqrt {13}$, and point $G$ is at distances $12\sqrt {3}$ and $36\sqrt {3}$ from vertices $A$ and $D$, respectively. Let $H$ be the foot of the altitude from $C$ to $\overline{AD}$. The distance $GH$ can be expressed in the form $p\sqrt {q}$, where $p$ and $q$ are positive integers and $q$ is not divisible by the square of any prime. Find $p + q$.

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