
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
Karl likes the number  $17$  his favorite polynomials are monic quadratics with integer coefficients such that  $17$  is a root of the quadratic and the roots differ by no more than  $17$ . Compute the sum of the coefficients of all of Karl's favorite polynomials. (A monic quadratic is a quadratic polynomial whose  $x^2$  term has a coefficient of  $1$ .)

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