
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 the function \( f(x)=4x + ax^2 - \frac{2}{3}x^3 \), where \( a \in [-1, 1] \) is a constant. Assume that the equation \( f(x) = 2x + \frac{1}{3}x^3 \) has two nonzero real roots \( x_1, x_2 \). Determine whether there exists a real number \( m \) such that the inequality \( m^2 + tm + 1 \geq \left| x_1 - x_2 \right| \) always holds for any \( a \in [-1, 1] \) and \( t \in [-1, 1] \). If such an \( m \) exists, find the range of \( m \); if not, explain why.

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