
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) If the coordinates of the focus of a parabola are $(2,0)$, then the standard equation of the parabola is __________.

(2) A sufficient but not necessary condition for "the line $x-y-k=0$ and the circle $(x-1)^{2}+y^{2}=2$ to have two distinct intersection points" can be __________.

(3) A hyperbola shares the same foci with the ellipse $\dfrac{x^{2}}{16}+\dfrac{y^{2}}{64}=1$, and one of its asymptotes is $y=x$. The equation of the hyperbola is __________.

(4) Given that the function $f(x)=x^{3}+ax^{2}+(a+6)x+1$ has no extreme values, the range of the real number $a$ is __________.

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