
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
Find the non-negative root and the corresponding value of \( x \) for the following cases:
1. When \( a \in (-\infty, 0) \):
   - \( t = \frac{1 - \sqrt{1 - 8a + 12a^2}}{2a} \)
   - \( x = \left( \frac{1 - \sqrt{1 - 8a + 12a^2}}{2a} \right)^2 \)

2. When \( a \in \left\{ \frac{1}{6}, \frac{1}{2} \right\} \cup \left( \frac{2}{3}, +\infty \right) \):
   - \( t = \frac{1 + \sqrt{1 - 8a + 12a^2}}{2a} \)
   - \( x = \left( \frac{1 + \sqrt{1 - 8a + 12a^2}}{2a} \right)^2 \)

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