
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 that positive numbers $x$ and $y$ satisfy $x+y=1$, find the minimum value of $\frac{1}{x}+\frac{4}{{1+y}}$;<br/>$(2)$ Find the minimum value of the function $y=\frac{{{x^2}+7x+10}}{{x+1}}(x \gt -1)$.

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