
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
An equilateral triangle $ABC$ has side length $a$. A smaller circle is inscribed within it, and point $P$ is any point inside the triangle. From $P$, perpendiculars $PD$, $PE$, and $PF$ are dropped to sides $AB$, $BC$, and $CA$ respectively. If point $P$ is the center of the circle with radius $r$ that touches each side of the triangle, what is the relation between the sum of the perpendicular distances and $a$ or $r$?
A) $\frac{\sqrt{3}a}{2}$
B) $\sqrt{3}a$
C) $\frac{a^2}{4}$
D) $\frac{a}{\sqrt{3}}$
E) $\frac{a^2}{\sqrt{3}}$

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