
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 $f(x) = ax\ln x + b$, where $a \neq 0$.  
$(1)$ If $a = 2$ and $b = 3$, find the equation of the tangent line to the curve $y = f(x)$ at the point $(1, f(1))$.  
$(2)$ Discuss the monotonicity of the function $f(x)$ and indicate the intervals of monotonicity of the function $f(x)$.

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