
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 fractional equation about $x$ is $\frac{mx-1}{x-2}=2+\frac{3}{2-x}$ has an integer solution, and the graph of the quadratic function $y=\left(m-2\right)x^{2}+2x+1$ intersects the $x$-axis, then the sum of all integers $m$ that satisfy the conditions is ______.

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