
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 $xOy$, with the origin as the pole and the positive $x$-axis as the polar axis, the parametric equation of curve $C_1$ is 
$$
\begin{cases} 
x=2+ \sqrt {3}\cos \theta \\ 
y= \sqrt {3}\sin \theta
\end{cases}
(\theta \text{ is the parameter}),
$$
and the polar equation of curve $C_2$ is $\theta= \frac {\pi}{6} (\rho \in \mathbb{R})$.

$(1)$ Find the general equation of curve $C_1$ and the Cartesian coordinate equation of curve $C_2$;

$(2)$ Curves $C_1$ and $C_2$ intersect at points $A$ and $B$. Given point $P(3, \sqrt {3})$, find the value of $||PA|-|PB||$.

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