
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 $f(x)=ax^{2}-(2a+1)x+2$, where $a\in \mathbb{R}$.  
$(1)$ When $a=0$, find the zeros of the function $f(x)$.  
$(2)$ When $a=1$, if the equation $f(x)=m$ has solutions for $x\in [1,3]$, find the range of real numbers $m$.  
$(3)$ When $a>0$, find the solution set of the inequality $f(x) > 0$ in terms of $x$.

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