
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 quadratic function $f(x) = x^2 - 2(m-1)x - 2m + m^2$,
(1) If its graph passes through the origin, find the value of $m$;
(2) If its graph is symmetric about the y-axis, write the expression for this function;
(3) Determine whether there exists a real number $m$, such that for every $x$ in the interval [1, 3], the inequality $f(x) \geq 3$ holds. If such an $m$ exists, find the range of $m$; if not, explain why.

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