
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 rectangular coordinate system $(xOy)$, a line $l_{1}$ is given by the equation $y = \tan \alpha \cdot x \ (0 \leqslant \alpha < \pi, \alpha \neq \frac{\pi}{2})$, and a parabola $C$ is given by the parametric equations $\begin{cases} x = t^{2} \\ y = -2t \end{cases} \ (t \text{ is a parameter})$. Establish a polar coordinate system with the origin $O$ as the pole and the non-negative semi-axis of the $x$-axis as the polar axis.

1. Find the polar equations of the line $l_{1}$ and the parabola $C$.
2. If the line $l_{1}$ intersects the parabola $C$ at point $A$ (distinct from the origin $O$), draw a line $l_{2}$ passing through the origin and perpendicular to $l_{1}$. The line $l_{2}$ intersects the parabola $C$ at point $B$ (distinct from the origin $O$). Find the minimum value of the area of triangle $OAB$.

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