
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
$(1)$ Given proposition $p:∃x>-1,\frac{x^{2}+3x+6}{x+1}<a$, if proposition $p$ is false, find the range of real number $a$. <br/> $(2)$ If positive numbers $a$ and $b$ satisfy $a+2b=1$, find the minimum value of $2a+\frac{1}{a}+4b+\frac{8}{b}$.

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