
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 $p:\forall x\in \left[-1,1\right]$, $x^{2}+2x-k\leqslant 0$, proposition $q:\exists x\in R$, $x^{2}+2kx+3k+4=0$.  
$(1)$ When proposition $\neg p$ is false, find the range of real number $k$;  
$(2)$ If propositions $p$ and $q$ have exactly one false proposition, find the range of real number $k$.

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