
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 real numbers $x \gt 0$, $y \gt 0$, and $2xy=x+y+a(x^{2}+y^{2})$, where $a\in R$.  
$(1)$ When $a=0$, find the minimum value of $2x+4y$ and determine the values of $x$ and $y$ that correspond to this minimum value.  
$(2)$ When $a=\frac{1}{2}$, find the range of values for $x+y$.

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