
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
Consider equations of the form $x^2 + bx + c = 0$. How many such equations have real roots, and have coefficients $b$ and $c$ selected from the set of integers $\{1, 2, 3, 7, 8, 9\}$, where $c$ is also an even number?

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