
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 that positive real numbers $x$ and $y$ satisfy $x+y=4$.  
$(1)$ Does there exist positive real numbers $x$ and $y$ such that $xy=5$? If so, find the values of $x$ and $y$; if not, explain why.  
$(2)$ Find the minimum value of $\frac{1}{{x+1}}+\frac{4}{y}$.

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