
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 $$f(x)=x+ \frac {b}{x}-3$$, where $x\in[1,2]$  
(1) If $b=1$, find the range of $f(x)$.  
(2) If $b\geq2$ and the maximum value of $f(x)$ is $M$, the minimum value is $m$, and it satisfies $M-m\geq4$, find the range of $b$.

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