
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 $f(x) = \ln(ax^2 + 2x + 1)$, $g(x) = \log_{\frac{1}{2}}(x^2 - 4x - 5)$.  
(1) If the domain of $f(x)$ is $\mathbb{R}$, find the range of the real number $a$.  
(2) If the range of $f(x)$ is $\mathbb{R}$, then find the range of the real number $a$.  
(3) Find the decreasing interval of the function $g(x)$.

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