
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
How many real triples \((a, b, c)\) are there such that the polynomial \(p(x) = x^4 + ax^3 + bx^2 + ax + c\) has exactly three distinct roots, which are equal to \(\tan y\), \(\tan 2y\), and \(\tan 3y\) for some real \(y\)?

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