
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 line $l$ is given by
$$
\begin{cases}
x = t\cos \frac{8\pi}{3} \\
y = -4 + t\sin \frac{8\pi}{3}
\end{cases}
(t \text{ is the parameter})
$$
With the origin as the pole and the positive half-axis of $x$ as the polar axis, a polar coordinate system is established. The polar equation of curve $C$ is:
$$\rho^2 - 3\rho - 4 = 0 (\rho \geqslant 0).$$
$(1)$ Write the general equation of line $l$ and the Cartesian coordinate equation of curve $C$;  
$(2)$ Suppose line $l$ intersects curve $C$ at points $A$ and $B$, find the value of $\angle AOB$.

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