
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 $xoy$, let the center of the circle $x^{2}+y^{2}-4x=0$ be $M$.
$(1)$ Find the equation of the line that passes through point $P(0, -4)$ and is tangent to circle $M$;
$(2)$ If a line passing through point $P(0, -4)$ with slope $k$ intersects circle $M$ at two distinct points $A$ and $B$, and the slopes of lines $OA$ and $OB$ are $k_{1}$ and $k_{2}$ respectively, is $k_{1}+k_{2}$ a constant? If so, find this constant; if not, explain why.

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