
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 $xOy$, curve $C_1$ is defined by the parametric equation
\[
\begin{cases}
x = \sqrt{m} + \frac{1}{\sqrt{m}}\\
y = \sqrt{m} - \frac{1}{\sqrt{m}}
\end{cases}
\]
where $m$ is the parameter. Line $l$ intersects curve $C_1$ at points $A$ and $B$. In the polar coordinate system with the origin as the pole and the positive x-axis as the polar axis, curve $C_2$ has the polar equation $\rho=4\sin(\theta- \frac{\pi}{6})$. Point $P(\rho, \frac{\pi}{3})$ lies on curve $C_2$.  
(1) Determine the Cartesian equation of curve $C_1$ and the Cartesian coordinates of point $P$;  
(2) If line $l$ has an angle of inclination $\frac{2\pi}{3}$ and passes through point $P$, find the value of $|PA| + |PB|$.

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