
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
In the Cartesian coordinate system, given the parabola $C: y = ax^{2} + 2x - 1 (a \neq 0)$ and the line $l: y = kx + b$, where the points $A(-2, -\frac{5}{2})$ and $B(3,0)$ lie on the line $l$. <br/>$(1)$ Find the expression of the line $l$; <br/>$(2)$ If the parabola $C$ intersects the line $l$, find the range of values for $a$; <br/>$(3)$ When $a = -1$, and the independent variable $x$ of the quadratic function $y = ax^{2} + 2x - 1$ satisfies $m \leq x \leq m+2$, and the maximum value of the function $y$ is $-4$, find the value of $m$.

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