
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 the parabola $y = x^2 - kx + k - 1$, find the value of $k$ under the following conditions:
1. The vertex of the parabola is on the x-axis;
2. The vertex of the parabola is on the y-axis;
3. The vertex of the parabola is at $(-1, -4)$;
4. The parabola passes through the origin;
5. The parabola has a minimum value of $y$ when $x = 1$;
6. The minimum value of $y$ is $-1$.

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