
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 $f(x) = 4^x + a \cdot 2^x + 3$, where $a \in \mathbb{R}$.

(1) When $a = -4$ and $x \in [0, 2]$, find the range of the function $f(x)$.

(2) If the equation $f(x) = 0$ has two different real roots in $(0, +\infty)$, find the range of the real number $a$.

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