
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
$ABCD$ is a rectangular piece of paper that has been folded so that corner $C$ meets point $C'$ on edge $AB.$ The crease is $GH,$ where $G$ is on $CD$ and $H$ is on $AB.$ Given the dimensions $DG=10, GC=23,$ and $HA=4$ are given. Find the perimeter of rectangle $ABCD$ in the simplest form $\frac{p}{q}$, where $p$ and $q$ are coprime positive integers, and calculate $p+q.$

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