
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 plane rectangular coordinate system $xOy$, the parameter equations of curve $C_{1}$ are $\left\{\begin{array}{l}x=\sqrt{3}\sin\alpha,\\ y=\sqrt{2}\cos\alpha\end{array}\right.$ ($\alpha$ is the parameter). Taking the coordinate origin $O$ as the pole and the non-negative half-axis of the $x$-axis as the polar axis, the polar coordinate equation of the line $l$ is $ρ({2\cos\theta+\sin\theta})=\sqrt{6}$, where $\rho\ \ \gt 0$, $0\leqslant \theta\ \ \lt 2\pi$.  
$(1)$ Find the Cartesian equation of curve $C_{1}$ and the equation of line $l$ in rectangular coordinates.  
$(2)$ Line $l$ intersects curve $C_{1}$ at points $A$ and $B$, and the polar angles corresponding to points $A$ and $B$ are $\theta _{1}$ and $\theta _{2}$, respectively. Find the value of $\theta _{1}+\theta _{2}$.

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