
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 point $Q(\sqrt{5},\frac{4}{3}$) is a point on the ellipse $C$: $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1(a>b>0)$, $F_{1}$, $F_{2}$ are the left and right foci of the ellipse, and $\overrightarrow{Q{F}_{2}}\cdot \overrightarrow{{F}_{2}{F}_{1}}=0$. Find the equation of the ellipse $C$______. If the tangent to the circle $x^{2}+y^{2}=4$ intersects the ellipse $C$ at points $M$ and $N$, then the maximum value of $|MN|$ is______.

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