
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
Rhombus $EFGH$ is inscribed in rectangle $JKLM$ such that vertices $E$, $F$, $G$, and $H$ lie on the sides $\overline{JK}$, $\overline{KL}$, $\overline{LM}$, and $\overline{MJ}$, respectively. It is given that $JE=12$, $EF=24$, $FK=16$, and $EG=32$. Let $p/q$, in lowest terms, denote the perimeter of rectangle $JKLM$. Find $p+q$.

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