
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
Let $f(x) = xe^{x}$.  
$(1)$ Discuss the monotonicity of the function $g(x) = f(x) - \frac{1}{2}a(x+1)^{2}$ where $a > 0$.  
$(2)$ Three tangent lines can be drawn to the curve $y = f(x)$ passing through the point $(1,m)$. Find the range of real number $m$.

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