
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 \( x_1 \) and \( x_2 \) be the real roots of the equation \( x^2 + ax + a - \frac{1}{2} = 0 \).

1. Find the set of possible values for the real number \( a \).
2. Is there a real number \( m \) such that the inequality \( m^2 + tm + 4 \sqrt{2} + 6 \geq (x_1 - 3x_2)(x_2 - 3x_1) \) holds for any \( a \) in the set found in (1) and for any \( t \in [-1, 1] \)? If such an \( m \) exists, determine the range of \( m \); if not, provide an explanation.

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