
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
The polynomial \( p(x) \equiv x^3 + ax^2 + bx + c \) has three positive real roots. Find a necessary and sufficient condition on \( a \), \( b \), and \( c \) for the roots to be \( \cos A \), \( \cos B \), and \( \cos C \) for some triangle \( ABC \).

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