
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 function $f(x)=3x-\frac{1}{x}+b\ln x$.<br/>$(1)$ When $b=-4$, find the local minimum value of the function $f(x)$;<br/>$(2)$ If there exists $x\in [1,e]$ such that $4x-\frac{1}{x}-f(x)＜-\frac{1+b}{x}$ holds, find the range of values for $b$.

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