
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 inequality $ax^{2}-2ax+3 \gt 0$ with respect to $x$.<br/>$(1)$ If the solution set of the inequality is $\{x\left|\right.-1 \lt x \lt 3\}$, find the real number $a$;<br/>$(2)$ If the solution set of the inequality is $R$, find the range of real numbers for $a$.

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