
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
Solve the following system of equations:

$$
\begin{aligned}
& x + y = a \\
& \frac{1}{x} + \frac{1}{y} = \frac{1}{b}
\end{aligned}
$$

Determine the values of \(a\) and \(b\) for which the roots are real. What relationship must exist between \(a\) and \(b\) for one root to be three times the other?

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