
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 $A$ and $B$ are polynomials, and $A=2x^{2}-mx+1$, $B=nx^{2}-3$.  
$(1)$ If the product of $A$ and $B$ does not contain the terms $x^{2}$ and $x^{3}$, find the values of $m$ and $n$;  
$(2)$ On the number line, let the point representing the number $m$ be denoted as $M$, and the point representing the number $n$ be denoted as $N$. Under the condition of $(1)$, the point $P$ on the number line satisfies that the distance from $P$ to point $M$ is twice the distance from $P$ to point $N$. Find the number represented by point $P$.

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