
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 polar equation of curve $C$ is $\rho ^{2}=\frac{4}{co{s}^{2}\theta+4si{n}^{2}\theta}$, with the pole as the origin $O$ of the Cartesian coordinate system and the polar axis as the non-negative x-axis.<br/>$(1)$ Find the Cartesian equation of curve $C$;<br/>$(2)$ Let $P$ and $Q$ be two points on curve $C$. If $OP\bot OQ$, find the value of $\frac{1}{|OP{|}^{2}}+\frac{1}{|OQ{|}^{2}}$.

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