
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
Let $f(x)=ax^{2}+(1-a)x-1$.  
$(1)$ Proposition $p$: There exists $x\in \mathbb{R}$ such that $f(x) < x-3$. If $P$ is a false proposition, find the range of real numbers $a$.  
$(2)$ Find the solution set of the inequality $f(x) < 0$ for $a < 0$.

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