
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 the function $f(x)=\frac{bx}{\ln x}-ax$, where $e$ is the base of the natural logarithm.

$(1)$ If the tangent line to the graph of $f(x)$ at the point $(e^2,f(e^2))$ is $3x+4y-e^2=0$, find the values of the real numbers $a$ and $b$;

$(2)$ When $b=1$, if there exist $x_1, x_2 \in [e,e^2]$ such that $f(x_1) \leqslant f'(x_2)+a$ holds, find the minimum value of the real number $a$.

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