
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 when the system will have two distinct solutions. The quadratic equation (*) will have two distinct non-negative roots \( x_{1,2} = \frac{3a \pm \sqrt{6-3a}}{3} \), if

\[
\left\{
\begin{array}{c}
6 - 3a > 0, \\
a > 0, \\
3a^2 + a - 2 \geq 0
\end{array}
\right.
\Leftrightarrow
\left\{
\begin{array}{l}
0 < a < 2, \\
\left[
\begin{array}{l}
a \leq -1, \\
a \geq \frac{2}{3}
\end{array}
\right.
\end{array}
\Leftrightarrow \frac{2}{3} \leq a < 2
\right.
\]

From this interval, we need to exclude 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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