
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
Let proposition p: the function $f(x) = x^2 - 2mx + 1$ is increasing on $[1, +\infty)$, and proposition q: the domain of the function $f(x) = \log(x^2 - mx + 1)$ is $\mathbb{R}$.

(1) If $m=2$, determine the truth of proposition p;  
(2) If one of propositions p and q is true and the other is false, find the range of the real number $m$.

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