
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
**Additional Question**: Given the quadratic function $y = x^2 + 2(m + 1)x - m + 1$.
1. As $m$ changes, does the vertex $P$ of this quadratic function always lie on a certain parabola? If yes, find the equation of this parabola; if no, explain why.
2. If the line $y = x + 1$ passes through the vertex $P$ of the quadratic function $y = x^2 + 2(m + 1)x - m + 1$, find the value of $m$ at this time.

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