
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 two pairs \((x,y)\) of real numbers that satisfy the equations \(x+y = 2xy = 5\). Given that the solutions \(x\) are in the form \(x = \frac{a \pm b\sqrt{c}}{d}\), where \(a\), \(b\), \(c\), and \(d\) are positive integers and the expression is completely simplified, what is the value of \(a + b + c + d\)?

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