
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 parabola $y=x^{2}+2bx+b^{2}-2\left(b \gt 0\right)$ is given.<br/>$(1)$ Express the coordinates of the vertex of the parabola in terms of $b$ as ______；<br/>$(2)$ If the parabola passes through the point $B\left(0,-1\right)$,<br/>① when $-2 \lt x \lt 3$, find the range of $y$；<br/>② if $k\leqslant x\leqslant 2$, $-2\leqslant y\leqslant 7$, combined with the graph of the function, find the range of $k$.

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