
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  $PQRS$  is inscribed in rectangle  $ABCD$  so that vertices  $P$ ,  $Q$ ,  $R$ , and  $S$  are interior points on sides  $\overline{AB}$ ,  $\overline{BC}$ ,  $\overline{CD}$ , and  $\overline{DA}$ , respectively. It is given that  $PB=15$ ,  $BQ=20$ ,  $PR=30$ , and  $QS=40$ . Let  $m/n$ , in lowest terms, denote the perimeter of  $ABCD$ . Find  $m+n$ .

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