
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 a circle $C$: $(x-a)^{2}+(y-a)^{2}=a^{2}(a > 0)$ with an area of $\pi$ and intersects the $x$-axis and $y$-axis at points $A$ and $B$ respectively.  
$(1)$ Find the equation of circle $C$;  
$(2)$ If the line $l$: $y=k(x+2)$ intersects with the segment $AB$, find the range of the real number $k$;  
$(3)$ Discuss the number of intersection points between the line $l$: $y=k(x+2)$ and the circle $C$: $(x-a)^{2}+(y-a)^{2}=a^{2}(a > 0)$.

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