
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 a constant $m\neq 0$, $n \geq 2$ and $n\in \mathbb{N}$, in the expansion of the binomial $(1+mx)^{n}$, only the coefficient of the sixth term is the maximum, and the coefficient of the third term is 9 times the coefficient of the second term.
$(1)$ Find the values of $m$ and $n$;
$(2)$ If we denote $(1+mx)^{n}=a_{0}+a_{1}(x+8)+a_{2}(x+8)^{2}+\ldots+a_{n}(x+8)^{n}$, find the remainder when $a_{0}-a_{1}+a_{2}-a_{3}+\ldots+(-1)^{n}a_{n}$ is divided by $6$.

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