
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
Given two points $A(x_1, y_1)$ and $B(x_2, y_2)$ on the parabola $y = 2x^2$ that are symmetric about the line $y = x + m$, and $x_1 \cdot x_2 = -\frac{1}{2}$, find the value of $m$.

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