
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
Suppose  $x$  and  $y$  are real numbers such that \[x^2+xy+y^2=2\qquad\text{and}\qquad x^2-y^2=\sqrt5.\] The sum of all possible distinct values of  $|x|$  can be written in the form  $\textstyle\sum_{i=1}^n\sqrt{a_i}$ , where each of the  $a_i$  is a rational number.  If  $\textstyle\sum_{i=1}^na_i=\frac mn$  where  $m$  and  $n$  are positive realtively prime integers, what is  $100m+n$ ?

*Proposed by David Altizio*

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