
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 $xOy$, given the curve ${C_1}:\left\{{\begin{array}{l}{x=\sqrt{2}\cos t}\\{y=\sin t}\end{array}}\right.$ (where $t$ is a parameter), and the curve $C_{2}$: $\rho =r\left(r \gt 0\right)$, with the coordinate origin $O$ as the pole, and the positive x-axis as the polar axis.<br/>$(1)$ Find the polar coordinate equation of curve $C_{1}$ and the Cartesian coordinate equation of curve $C_{2}$;<br/>$(2)$ Given that $A$ and $B$ are two moving points on curve $C_{1}$ (other than the origin), and $\angle AOB=90^{\circ}$, if curve $C_{2}$ has exactly one point in common with the line $AB$, find the value of $r$.

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