
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 $f(x)=ax-\ln x$ and $F(x)=e^{x}+ax$, where $x > 0$ and $a < 0$.

$(1)$ If $f(x)$ and $F(x)$ have the same monotonicity on the interval $(0,\ln 3)$, find the range of the real number $a$.

$(2)$ If $a\in(-\infty,- \frac {1}{e^{2}}]$ and the function $g(x)=xe^{ax-1}-2ax+f(x)$ has a minimum value of $\varphi(a)$, find the minimum value of $\varphi(a)$.

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