
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 the function $f(x) = (x-a)^2 + be^{x}$, where $a, b \in \mathbb{R}$.  
$(1)$ If $a=0$, the function $y=f(x)$ has $2$ extreme points. Find the range of values for $b$.  
$(2)$ If $a=1$ and $b=\frac{2}{e}$, how many solutions does the equation $f(x)=3$ have?

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