
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
From the numbers $-1$, $-\frac{2}{3}$, $-\frac{1}{2}$, $0$, $1$, randomly select a number, denoted as $a$. Let $a$ be the number that makes the solution of the equation in terms of $x$: $\frac{ax}{x-3}-\frac{2}{3-x}=1$ a non-negative number. Also, the probability that the inequality system for $x$: $\left\{\begin{array}{l}{x-a>0}\\{-3+2x\leq 1}\end{array}\right.$ has three integer solutions is ____.

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