
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
Determine for which real values of \( m \) the simultaneous equations 
\[
\begin{align*}
y &= mx + 3, \\
y &= (3m - 2)x^2 + 5
\end{align*}
\]
have at least one pair of real numbers \( (x,y) \).
A) \( m \leq 12 - 8\sqrt{2} \ \text{or} \ m \geq 12 + 8\sqrt{2} \) \\
B) \( m \geq 12 - 8\sqrt{2} \ \text{and} \ m \leq 12 + 8\sqrt{2} \) \\
C) \( m = 12 \) \\
D) All real \( m \) \\
E) No values of \( m \)

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