
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 proposition $p$: $\exists x \in \mathbb{R}$, $kx^{2}+1 \leqslant 0$, and proposition $q$: $\forall x \in \mathbb{R}$, $x^{2}+2kx+1 > 0$.
$(1)$ When $k=3$, write the negation of proposition $p$ and judge its truthfulness;
$(2)$ When $p \lor q$ is a false proposition, find the range of real numbers $k$.

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