
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 $f(x) = 2^x|2^x-a|+2^{x+1}-3$, where $a$ is a real number.  
(1) If $a=4$ and $x \in [1, 3]$, find the range of $f(x)$.  
(2) If $f(x)$ is monotonic on $\mathbb{R}$, find the range of $a$.

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