
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 $f(x)=\ln x+ax$ has a tangent line at point $(t,f(t))$ with the equation $y=3x+1$.

$(1)$ Find the value of $a$;

$(2)$ Given $k\leqslant 2$, when $x > 1$, $f(x) > k(1-\dfrac{3}{x})+2x-1$ always holds, find the range of the real number $k$;

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