
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
Let \( x \) and \( y \) be real numbers such that
\[ 3x^2 + 6xy + 4y^2 = 1. \]
Let \( m \) and \( M \) be the minimum and maximum values of \( x^2 + 2xy + y^2 \), respectively. Find the product \( mM \).

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