
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
Find, with proof, all triples of real numbers  $(a, b, c)$  such that all four roots of the polynomial  $f(x) = x^4 +ax^3 +bx^2 +cx+b$  are positive integers. (The four roots need not be distinct.)

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