
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
Given $a \gt 1$, the solution set $M$ of the inequality $[\left(a-1\right)x-a^{2}+4a-6]\cdot \left(x+3\right) \lt 0$ with respect to $x$ is defined as $M$. Let $N=M\cap Z$. As $a$ varies, the set $N$ with the minimum number of elements in $N$ is ____.

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