
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 the point $M\left(\frac{2\sqrt{6}}{3},\frac{2}{3}\right)$ lies on the ellipse $C_{1}$: $\frac{{y}^{2}}{{a}^{2}}+\frac{{x}^{2}}{3}=1(a>0)$, and the upper focus of the ellipse coincides with the focus of the parabola $C_{2}$: $x^{2}=2py(p>0)$. A moving point $Q$ on the ellipse $C_{1}$ is used to draw two tangents to the parabola $C_{2}$, with the points of tangency being $A$ and $B$. <br/>$(1)$ Find the equations of $C_{1}$ and $C_{2}$; <br/>$(2)$ When point $Q$ moves on the curve below the $x$-axis of the ellipse $C_{1}$, find the maximum area of $\triangle QAB$.

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