
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 $XOY$, the equation of circle $C$ is $x^{2}+y^{2}-8x+15=0$. If there is at least one point on the line $y=kx-2$ such that a circle with this point as its center and a radius of $1$ has a common point with circle $C$, then the maximum value of $k$ is ______.

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