
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 a straight line $l$ passing through the origin with an inclination angle of $\theta_0$ ($\theta_0 \neq \frac{\pi}{2}$), establish a polar coordinate system with the origin $O$ as the pole and the positive semi-axis of the $x$-axis as the polar axis. The polar equation of curve $C$ is given by $\rho \sin^2 \theta = 4 \cos \theta$.
$(I)$ Write the polar equation of the line $l$ and the Cartesian equation of the curve $C$;  
$(II)$ A straight line $l'$ passes through the origin and is perpendicular to line $l$. If $l \cap C = M$ and $l' \cap C = N$ where $M$ and $N$ do not coincide with the origin, find the minimum value of the area of the triangle $\triangle OMN$.

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