
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
In a pocket, there are six small balls that are identical in size and shape, each labeled with one of the numbers $-5, -4, -3, -2, 2, 1$. We randomly draw one ball from the pocket and denote the number on it as $a$. Determine the probability that the vertex of the parabola given by \( y = x^2 + 2x + a + 2 \) falls in the third quadrant, and the equation
\[
\frac{ax}{x-2} = \frac{3x+2}{2-x} + 2
\]
has an integer solution.

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