
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 $f(x)=ae^{x}$, where $a\in \mathbb{R}$.  
$(1)$ If the tangent line of the graph of $y=f(x)$ at the point $(1,f(1))$ passes through the point $(3,3)$, find the monotonic interval of the function $y=xf(x)$.  
$(2)$ Let $g(x)=x^{2}$, and $a>0$. If the curve $y=f(x)$ and the curve $y=g(x)$ have a unique common tangent line, find the value of the real number $a$.

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