
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
The parabola $y=ax^2+\frac{1}{2}x+c$ intersects the $x$-axis at points $A$ and $B$, and intersects the $y$-axis at point $C$. The line $y=-\frac{1}{2}x-2$ passes through points $A$ and $C$. <br/>$(1)$① Find the values of $a$ and $c$; <br/>② Let the vertex of the parabola $y=ax^2+\frac{1}{2}x+c$ be $D$, find the area of $\triangle ACD$. <br/>$(2)$ Through point $M\left(m,0\right)\left(-4 \lt m \lt 0\right)$, draw a line perpendicular to the $x$-axis intersecting the parabola $y=ax^2+\frac{1}{2}x+c$ at point $N$, find the maximum value of the line segment $OM+MN$.

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