
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 in terms of $x$: $(k^{2}+2k-3)x^{2}+(k+3)x-1 \gt 0$ where $k\in R$, and the solution set of this inequality is denoted as $M$.  
$(1)$ If $M=\varnothing$, find the range of real numbers for $k$;  
$(2)$ If there exist two distinct positive real numbers $a$ and $b$ such that $M=(a,b)$, find the range of real numbers for $k$.

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