
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 $f(x) = \ln x$ and $g(x) = \frac{a}{x}$ ($a > 0$), let $F(x) = f(x) + g(x)$.

$(1)$ Find the intervals of monotonicity for the function $F(x)$;

$(2)$ Suppose point $P(x_{0}, y_{0})$ is any point on the graph of $y=F(x)$, and when $x_{0} \in (0,3]$, the slope of the tangent line at point $P$, denoted as $k$, always satisfies $k \leqslant \frac{1}{2}$. Find the minimum value of the real number $a$;

$(3)$ Is there a real number $m$ such that the graph of the function $y = g(\frac{2a}{x^{2}+1}) + m - 1$ has exactly four distinct intersection points with the graph of $y = f(1 + x^{2})$? If it exists, find the range of the real number $m$; if not, explain why.

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