
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 functions $f(x) = \lg(kx)$, $g(x) = \lg(x+1)$, and $h(x) = \frac{x}{x^2+1}$.
1. When $k=1$, find the monotonic interval of the function $y=f(x)+g(x)$;
2. If the equation $f(x)=2g(x)$ has only one real root, find the set of values for the real number $k$;
3. Let $p(x)=h(x)+\frac{mx}{1+x}$ have exactly two distinct zeros in the interval $(-1,1)$. Find the range of values for the real number $m$.

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