
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 $f(x) = x^2 + (4a-3)x + 3a$  
(1) When $a=1$ and $x \in [-1, 1]$, find the range of the function $f(x)$.  
(2) Given $a > 0$ and $a \neq 1$, if the function $g(x) = \begin{cases} f(x), & x < 0 \\ \log_{a}(x+1)+1, & x \geq 0 \end{cases}$ is a decreasing function on $\mathbb{R}$, find the range of the real number $a$.

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