
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 a quadratic function $f\left(x\right)=-x^{2}+2ax-3$.  
$(1)$ When $a=-2$, the inequality $f\left(x\right) \lt 0$ has solutions.  
$(2)$ If $x\in \left[1,5\right]$, the inequality $f\left(x\right) \lt 3ax$ always holds. Find the range of real numbers.

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