
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 functions $f(x)=ax\ln x$ $(a\neq 0)$ and $g(x)=kx-1$.  
$(1)$ Find the monotonic intervals of $f(x)$.  
$(2)$ When $a=1$, if the graphs of $f(x)$ and $g(x)$ have two distinct intersection points in the interval $[\frac{1}{e},e]$, find the range of values for $k$.

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