
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
The equation (*) has one non-negative root \( x = 2a + 2 \sqrt{a^2 - 3a + 2} \).

For \( x < 0 \), \( x = \frac{3a - 2}{a} < 0 \Rightarrow 0 < a < \frac{2}{3} \).

Comparing this with [ unspecified condition ] I, we notice that for \( 0 < a < \frac{2}{3} \) there will also be two distinct roots.

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