
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 $f(x)=\frac{m}{2}x^2+(m-1)x-1$ where $m\in \mathbb{R}$.  
$(1)$ Find the maximum value of the function $f(x)$ on the interval $[1,2]$.  
$(2)$ If $m$ is an integer and the inequality $f(x)\geq \ln x$ holds for all $x$, find the minimum value of the integer $m$.

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