
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
Let  $f(x)=x^2+x$  for all real  $x$ . There exist positive integers  $m$  and  $n$ , and distinct nonzero real numbers  $y$  and  $z$ , such that  $f(y)=f(z)=m+\sqrt{n}$  and  $f(\frac{1}{y})+f(\frac{1}{z})=\frac{1}{10}$ . Compute  $100m+n$ .

*Proposed by Luke Robitaille*

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