
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
The function $f(x)=\lg (x^{2}+ax-a-1)$, among the following statements, the true statement is __________.

$(1)$ When $a=-2$, the domain of the function is $\{x|x\neq -1\}$,

$(2)$ $f(x)$ has a minimum value, but no maximum value;

$(3)$ When $a=0$, the range of the function is $\mathbb{R}$,

$(4)$ When $a\geqslant -2$, the function is increasing on $[1,+\infty)$,

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