
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, the graph of the quadratic function $y=ax^{2}+bx+4\left(a \lt 0\right)$ intersects the $x$-axis at points $A\left(-2,0\right)$ and $B\left(4,0\right)$, and intersects the $y$-axis at point $C$. The line $BC$ intersects the axis of symmetry at point $D$.  
$(1)$ Find the analytical expression of the quadratic function.  
$(2)$ If there is a point $M$ on the axis of symmetry of the parabola $y=ax^{2}+bx+4\left(a \lt 0\right)$, and the quadrilateral with vertices $O$, $C$, $D$, and $M$ is a parallelogram, find the coordinates of point $M$.

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