
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 that the fractional equation about $x$ is $\frac{{mx-1}}{{x-1}}=2+\frac{1}{{1-x}}$ has an integer solution, and the graph of the quadratic function $y=\left(m-1\right)x^{2}+2x+\frac{1}{2}$ intersects the $x$-axis, calculate the number of integers $m$ that satisfy the conditions.

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