
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 $y$ is a quadratic function of $x$ defined as $y=2mx^{2}+\left(1-m\right)x-1-m$, which of the following statements is correct?  
① When $m=-1$, the coordinates of the vertex of the graph are $(\frac{1}{2},\frac{1}{2})$;  
② When $m \gt 0$, the length of the segment intercepted by the graph on the $x$-axis is greater than $\frac{3}{2}$;  
③ When $m\neq 0$, the graph always passes through the fixed points;  
④ If two different points $P_{1}(x_{1},y_{1})$ and $P_{2}(x_{2},y_{2})$ are taken on the graph, then when $m \lt 0$, the inequality $\frac{{y}_{2}-{y}_{1}}{{x}_{2}-{x}_{1}} \lt 0$ must hold for $x \gt \frac{1}{4}$.

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