
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 $a < 0$, with $e$ being the base of the natural logarithm.
(Ⅰ) If $f(x)$ and $F(x)$ have the same monotonicity in the interval $(0,\ln 3)$, find the range of the real number $a$.
(Ⅱ) If $a \in \left( -\infty, -\frac{1}{e^2} \right]$, and the function $g(x) = xe^{ax-1} - 2ax + f(x)$ has the minimum value $M$, find the minimum possible value of $M$.

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