
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
All points of segment \( AB \) are interior points of an equilateral triangle. Project \( AB \) perpendicularly onto each side of the triangle. In which direction of \( AB \) will the sum of the projections be the largest, and in which direction will it be the smallest?

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