
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
For the function $f(x) = \ln x + x - a + \frac{b}{x}$, for any $x \in (0, +\infty)$, it holds that $f(x) \geqslant 0$ always.  
$(1)$ When $a = 3$, find the range of values for $b$.  
$(2)$ When $a > 0$, if for all $a$ that satisfy the condition, there exists $b$ such that $\frac{a}{b+1} \leqslant M$ always holds, find the minimum value of $M$.

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