
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
The function $y=\lg(3-4x+x^2)$ has a domain of $M$, and the function $f(x)=4^x-2^{x+1}$, where $x \in M$.  
(1) Find $M$;  
(2) Find the range of the function $f(x)$;  
(3) When $x \in M$, if the equation $4^x-2^{x+1}=b$ ($b \in \mathbb{R}$) has real roots for $x$, find the range of $b$ and discuss the number of real roots.

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