
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 $xOy$, a line $l$ passes through the point $P(-3,0)$, with an angle of inclination $\alpha$. Using the origin $O$ as the pole, the non-negative half-axis of the $x$-axis as the polar axis, and taking the same length units as the Cartesian coordinate system $xOy$, we establish the polar coordinate system. The polar equation of the curve $C$ is $\rho^{2} - 2\rho\cos\theta - 3 = 0$.

$(1)$ If line $l$ has a common point with curve $C$, determine the range of the angle of inclination $\alpha$.

$(2)$ Let $M(x,y)$ be any point on curve $C$, determine the range of $x+y$.

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