
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
Consider cases where the system has two distinct solutions. The quadratic equation (*) will have two distinct non-negative roots \( x_{1,2} = \frac{3a \pm \sqrt{6-3a}}{3} \) if:
\[ \begin{cases} 
6 - 3a > 0, \\ 
a > 0, \\ 
3a^{2} + a - 2 \geq 0 
\end{cases} \Leftrightarrow 
\begin{cases} 
0 < a < 2, \\ 
\begin{cases} 
a \leq -1, \\ 
a \geq \frac{2}{3}
\end{cases} 
\end{cases} \Leftrightarrow \frac{2}{3} \leq a < 2.
\]
From this interval, we need to remove the previously considered point \( a = \frac{5}{3} \). Combining the found values of \( a \), we get the answer.

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