
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
If $a$ makes the system of inequalities about $x$ $\left\{\begin{array}{c}6x-5≥a\\ \frac{x}{4}-\frac{x-1}{6}＜\frac{1}{2}\end{array}\right.$ have exactly $2$ integer solutions, and also makes the equation about $y$ $4y-3a=2\left(y-3\right)$ have a positive solution, then the number of all integers $a$ satisfying the conditions is ______.

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