
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
$A B C$ is an equilateral triangle with center $O$. Let $f$ be a line passing through $O$ and intersecting the sides of the triangle at points $A_{1}, B_{1}, C_{1}$. What is the angle between $f$ and the sides of the triangle if the product $O A_{1} \cdot O B_{1} \cdot O C_{1}$ is minimized?

What are the angles if the product is twice the minimum value?

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