
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
Let \( ABC \) be an equilateral triangle. Through lines \( AB \), \( BC \), and \( AC \), there are three planes that form an angle \( \phi \) with the plane \( ABC \) and intersect at point \( D_1 \). Additionally, through these lines, there are planes forming an angle \( 2\phi \) with the plane \( ABC \) and intersecting at point \( D_2 \). Find \( \phi \) given that points \( D_1 \) and \( D_2 \) are at equal distances from the plane \( ABC \).

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