
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
A point is randomly chosen inside an equilateral triangle of side length 10 units. Perpendiculars are dropped from this point to each of the three sides. Find the sum of the lengths of these perpendiculars.
A) $15$
B) $10\sqrt{3}$
C) $5\sqrt{3}$
D) $25$
E) $30\sqrt{3}$

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