
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
Set the origin as the pole and the positive half-axis of the $x$-axis as the polar axis to establish a polar coordinate system. The parametric equation of curve $C$ is given by: 
$$
\begin{cases} 
x= \sqrt {3}\cos \phi \\ 
y=\sin \phi
\end{cases}
(\phi \text{ is the parameter}),
$$
and the polar equation of line $l$ is $\rho(\cos \theta+\sin \theta)=4$.

$(1)$ Find the general equation of curve $C$ and the Cartesian coordinate equation of line $l$;  
$(2)$ If point $P$ is on curve $C$, and point $Q$ is on line $l$, find the minimum value of the length of segment $PQ$.

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