
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 positive integers  $m$  and  $n$  such that  $m^2 -n = 32$  and   $\sqrt[5]{m +\sqrt{n}}+  \sqrt[5]{m -\sqrt{n}}$  is a real root of the polynomial  $x^5 - 10x^3 + 20x - 40$ . Find  $m + n$ .

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