
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 $IJKL$ so that vertices $E$, $F$, $G$, and $H$ are interior points on sides $\overline{IJ}$, $\overline{JK}$, $\overline{KL}$, and $\overline{LI}$, respectively. It is given that $IE=12$, $EJ=25$, $EG=35$, and $FH=42$. Let $m/n$, in lowest terms, denote the perimeter of $IJKL$. Find $m+n$.

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