
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
In the interval $D$, if the function $f(x)$ is a decreasing function, and $xf(x)$ is an increasing function, then $f(x)$ is called a weakly decreasing function on $D$. Suppose $f(x) = \frac {1}{ \sqrt {1+x}}$  
(1) Determine whether $f(x)$ is a weakly decreasing function on the interval $[0, +\infty)$;  
(2) When $x \in [1, 3]$, the inequality $\frac {a}{x} \leq \frac {1}{ \sqrt {1+x}} \leq \frac {a+4}{2x}$ always holds, find the range of the real number $a$;  
(3) If the function $g(x) = f(x) + k|x| - 1$ has two different zeros on $[0, 3]$, find the range of the real number $k$.

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