
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 $18$. 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 $AC.$ If $PQ = 4$ and $RS = 10$, compute the square of the length of $\overline{BD}$, denoted as $d^2$. Assume $d$ represents the longer diagonal of $ABCD$.
A) 176  
B) 180  
C) 196  
D) 208  
E) 225

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