
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 $f(x)=ax^{2}-(2a+1)x-1$ where $a\in \mathbb{R}$.  
$(1)$ For all $x\in \mathbb{R}$, if $f(x)≤-\frac{3}{4}$, find the range of $a$.  
$(2)$ If $a\leqslant 0$ and for all $x \gt 0$, $xf(x)≤1$, find the range of $a$.

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