
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 $f(x)=x\cdot e^{x}$, if the equation regarding $x$, $\left[f(x)+ \frac{1}{2e}\right]\cdot\left[f(x)+\lambda\right]=0$ has exactly $3$ distinct real solutions, then the range of the real number $\lambda$ is __________.

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