
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 curve $C_1$ is $\begin{cases} x= \sqrt {m}+ \dfrac {1}{ \sqrt {m}} \\ y= \sqrt {m}- \dfrac {1}{ \sqrt {m}} \end{cases}$ (where $m$ is the parameter), and line $l$ intersects curve $C_1$ at points $A$ and $B$; with the origin as the pole and the positive half-axis of $x$ as the polar axis, the polar equation of curve $C_2$ is $\rho=4\sin (\theta- \dfrac {\pi}{6})$, and point $P(\rho, \dfrac {\pi}{3})$ is on curve $C_2$.
$(1)$ Find the general equation of curve $C_1$ and the Cartesian coordinates of point $P$;
$(2)$ If the slope angle of line $l$ is $\dfrac {2\pi}{3}$ and it passes through point $P$, find the value of $|PA|+|PB|$.

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