
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
Find the range of real numbers $a$ such that the inequality $a x^{2}-2 a x+2 a-3<0$ satisfies the following conditions respectively:
(1) The solution set is $\mathbf{R}$ (the set of all real numbers).
(2) The solution set is $\varnothing$ (the empty set).
(3) There is at least one real solution.

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