
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 that $x_1$ and $x_2$ are the two real roots of the quadratic equation in $x$, $x^2 - 2(m-1)x + m + 1 = 0$, and let $y = x_1^2 + x_2^2$.  
(1) Find the explicit expression for $y$ as a function of $m$, denoted as $y=f(m)$.  
(2) Determine the range of $y=f(m)$.

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