
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 a parallelogram with area $24$. Points $P$ and $Q$ are the projections of $A$ and $C,$ respectively, onto the line $BD;$ and points $R$ and $S$ are the projections of $B$ and $D,$ respectively, onto the line $AC.$ Suppose $PQ=8$ and $RS=10,$ and let $d$ denote the length of $\overline{BD},$ the longer diagonal of $ABCD.$ Calculate $d^2$ in the form $m+n\sqrt{p},$ where $m,n,$ and $p$ are positive integers and $p$ is not divisible by the square of any prime.
A) $93 + 16 \sqrt{24}$
B) $96 + 18 \sqrt{41}$
C) $100 + 20 \sqrt{33}$
D) $104 + 22 \sqrt{29}$

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