
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 $x$ and $y$ are both positive numbers.

$(1)$ If $x$ and $y$ satisfy $x+4y+xy=12$, find the maximum value of $xy$, and the values of $x$ and $y$ at this point.

$(2)$ If $x$ and $y$ satisfy $x+4y=xy$, find the minimum value of $x+y$.

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