
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
Define: If the function $y=f(x)$ exists a real number $x_{0}$ in its domain, such that $f(x_{0}+k)=f(x_{0})+f(k)$ holds, where $k$ is a constant greater than $0$, then the point $(x_{0}, k)$ is called the $k$-th "translation point" of the function $f(x)$.

$(1)$ Determine the number of $2$nd "translation points" of the function $g(x)=x\ln(x+1)$ and find the $2$nd "translation point";
$(2)$ If the function $h(x)=ax^{2}+x\ln x$ has a $1$st "translation point" on $[1,+\infty)$, find the range of real number $a$.

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