
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
Given that the equation of circle $O$ is $x^{2}+y^{2}=13$, and line $l$: $x_{0}x+y_{0}y=13$, let point $A(x_{0},y_{0})$.
$(1)$ If point $A$ is outside circle $O$, determine the positional relationship between line $l$ and circle $O$;
$(2)$ If point $A$ is on circle $O$, and $x_{0}=2$, $y_{0} > 0$, draw lines $AM$ and $AN$ passing through point $A$ intersecting circle $O$ at points $M$ and $N$ respectively, and the slopes of lines $AM$ and $AN$ are opposite numbers.
$\quad\quad①$ If line $AM$ passes through point $O$, find the value of $\tan ∠MAN$;
$\quad\quad②$ Discuss whether the slope of line $MN$ is a constant value regardless of how the slope of line $AM$ changes. If it is, find this constant value; if not, explain the reason.

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