
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]


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Given that the constant $m\neq 0$, $n\geqslant 2$ and $n\in N$, the expansion of the binomial ${{(1+mx)}^{n}}$ has only the 6th term with the maximum binomial coefficient, and the coefficient of the 3rd term is 9 times that of the 2nd term.

(1) Find the values of $m$ and $n$;
(2) If ${{(1+mx)}^{n}}={{a}\_{0}}+{{a}\_{1}}(x+8)+{{a}\_{2}}{{(x+8)}^{2}}+{{a}\_{3}}{{(x+8)}^{3}}+\cdots +{{a}\_{n}}{{(x+8)}^{n}}$, find the remainder when ${{a}\_{0}}-{{a}\_{1}}+{{a}\_{2}}-{{a}\_{3}}+\cdots +{{(-1)}^{n}}{{a}\_{n}}$ is divided by $6$.

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