
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 that the polar equation of curve $C$ is $\rho=2\cos \theta$, with the pole being the origin of the Cartesian coordinate system and the polar axis being the positive $x$-axis, set up the Cartesian coordinate system. The parametric equation of line $l$ is 
$$
\begin{cases}
x= \dfrac { \sqrt {3}}{2}t+m \\
y= \dfrac {1}{2}t
\end{cases}
(t \text{ is the parameter}).
$$
$(1)$ Find the Cartesian equation of curve $C$ and the general equation of line $l$;  
$(2)$ When $m=2$, line $l$ intersects curve $C$ at points $A$ and $B$. Find the value of $|AB|$.

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