
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 that the solution set of the inequality $x^{2}-3x+m \lt 0$ with respect to $x$ is $\{x\left|\right.1 \lt x \lt n\}$.  
$(1)$ Find the values of the real numbers $m$ and $n$;  
$(2)$ If positive numbers $a$ and $b$ satisfy: $ma+2nb=3$, find the maximum value of $a\cdot b$.

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