
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
(1) The equation of the tangent line to the function $y=x^{2}+x-1$ at the point $(1,1)$ is ______________.

(2) Given the ellipse $\dfrac{x^{2}}{36}+ \dfrac{y^{2}}{9}=1$ and a point inside the ellipse $P(4,2)$, the slope of the line containing the chord with $P$ as its midpoint is ______________.

(3) Given the sets $A=\{ x| 2x^{2}-x-3 < 0\}$ and $B=\{ x| y=\lg \dfrac{1-x}{x+3} \}$, when a real number $x$ is randomly selected from the interval $(-3,3)$, the probability that $x \in (A\cap B)$ is ______________.

(4) In the Cartesian coordinate system $XOY$, the equation of circle $C$ is $x^{2}+y^{2}-8x+15=0$. If there exists 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 common points with circle $C$, then the maximum value of $k$ is ______________.

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