
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
Let $f(x) = \ln x + 1$, $g(x) = ax + 2$, where $a \in \mathbb{R}$. Define $F(x) = f(x) - g(x)$.  
$(1)$ Find the equation of the tangent line to the curve $y = f(x)$ at $x = 1$.  
$(2)$ Determine the intervals where the function $F(x)$ is monotonic.  
$(3)$ If the graph of the function $f(x) = \ln x + 1$ always lies below the graph of $g(x) = ax + 2$, find the range of real numbers for $a$.

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