
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 function $f(x) = x^2 - 2mx + 2 - m$.

(I) If the inequality $f(x) \geq x - mx$ holds true for all $x$ in $\mathbb{R}$, find the range of the real number $m$.

(II) Let $A = \{y | y = f(x), 0 \leq x \leq 1\}$, and $A \subseteq [0, +\infty)$, find the maximum value of the real number $m$.

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