
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
Consider a square ABCD with side length $s$. A point P is chosen as the midpoint of side AB. From point P, perpendiculars are dropped to side CD and the line through AC (the diagonal of the square). What is the sum of the lengths of these perpendicular drops?
A) $s$
B) $s\left(1 + \frac{\sqrt{2}}{4}\right)$
C) $\frac{s}{\sqrt{2}}$
D) $\frac{s}{2} + \frac{s}{2\sqrt{2}}$

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