
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
There are real numbers  $a, b, c, $  and  $d$  such that  $-20$  is a root of  $x^3 + ax + b$  and  $-21$  is a root of  $x^3 + cx^2 + d.$  These two polynomials share a complex root  $m + \sqrt{n} \cdot i, $  where  $m$  and  $n$  are positive integers and  $i = \sqrt{-1}.$  Find  $m+n.$ 

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