
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 the one-variable linear equation $\frac{ax-1}{2}=7$ has a positive integer solution for $x$, and the system of inequalities about $x$ $\left\{\begin{array}{l}{2x-a\geq 0}\\{\frac{x-2}{2}<\frac{x+3}{3}}\end{array}\right.$ has at least $4$ integer solutions, determine the number of integers $a$ that satisfy all conditions.

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