
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 equation of curve $C$ is given by
$$
\begin{cases}
x = \sqrt{2}\cos\varphi \\
y = \sin\varphi
\end{cases}
(\varphi\ \text{is the parameter}).
$$
Establish a polar coordinate system with the origin as the pole and the positive half-axis of $x$ as the polar axis. Let $A$ and $B$ be two points on curve $C$ such that $OA \perp OB$, where $OA$ lies on the ray with $\theta = \alpha$ and $0 < \alpha < \frac{\pi}{2}$.

$(1)$ Find the polar equation of curve $C$.

$(2)$ Find the minimum value of $|OA| \cdot |OB|$.

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