
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 $f(x)=x\ln x+mx$, and the slope of the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$ is $1$.
$(1)$ Find the value of the real number $m$;
$(2)$ Let $g(x)=f(x)- \frac {a}{2}x^{2}-x+a$ $(a\in\mathbb{R})$ have two different extreme points $x_{1}$, $x_{2}$ in its domain, with $x_{1} < x_{2}$. Given $\lambda > 0$, if the inequality $e^{1+\lambda} < x_{1}\cdot x_{2}^{\lambda}$ always holds, find the range of $\lambda$.

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