
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 equation \(\frac{6-2a}{a}=0\) and the inequality \(\frac{4}{a}<0\), there are no such \(a\).

Determine in which intervals of \(a\) the equation \((*)\) has exactly one non-negative root given by:
1. For \(a \in (-\infty, 0)\), \(t = \frac{2 - \sqrt{4 - 6a + 2a^2}}{a}\) and \(x = \left(\frac{2 - \sqrt{4 - 6a + 2a^2}}{a}\right)^2\).
2. For \(a \in \{1, 2\} \cup (3, +\infty)\), \(t = \frac{2 + \sqrt{4 - 6a + 2a^2}}{a}\) and \(x = \left(\frac{2 + \sqrt{4 - 6a + 2a^2}}{a}\right)^2\).

Combine these cases to find the complete solution.

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