
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 functions $f(x)= \frac {\ln x}{x}$, $g(x)=kx(k > 0)$, and the function $F(x)=\max\{f(x),g(x)\}$, where $\max\{a,b\}= \begin{cases} a, & \text{if } a\geqslant b\\ b, & \text{if } a < b \end{cases}$  
$(I)$ Find the extreme value of $f(x)$;  
$(2)$ Find the maximum value of $F(x)$ on the interval $[1,e]$ ($e$ is the base of the natural logarithm).

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