
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 $xOy$, the parametric equations of curve $C_{1}$ are $\left\{{\begin{array}{l}{x=2\cos\varphi}\\{y=\sqrt{2}\sin\varphi}\end{array}}\right.$ (where $\varphi$ is the parameter). Taking point $O$ as the pole and the positive half-axis of the $x$-axis as the polar axis, the polar coordinate equation of curve $C_{2}$ is $\rho \cos^{2}\theta +4\cos\theta -\rho =0$.  
$(1)$ Find the general equation of curve $C_{1}$ and the Cartesian equation of curve $C_{2}$.  
$(2)$ The ray $l: \theta =\alpha$ intersects curves $C_{1}$ and $C_{2}$ at points $A$ and $B$ (both different from the pole). When $\frac{\pi}{4} \leq \alpha \leq \frac{\pi}{3}$, find the minimum value of $\frac{{|{OB}|}}{{|{OA}|}}$.

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