
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 inequality system about x is $\left\{\begin{array}{l}{x+m≤4}\\{\frac{x}{2}-\frac{x-1}{4}＞1}\end{array}\right.$ has a solution and at most $4$ integer solutions, and the polynomial $x^{2}-\left(1-m\right)$ can be factored within the range of rational numbers, determine the number of integers $m$ that satisfy the conditions.

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