
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 $f(x)=\log_{3}(a{x}^{2}+3x+a+\frac{5}{4})(a∈R)$.  
$(1)$ If the domain of $f(x)$ is $R$, find the range of real number $a$;  
$(2)$ If $f(x)$ is monotonically increasing on $(-\frac{1}{4},-\frac{1}{8})$, find the range of real number $a$.

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