
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 sixth term has the maximum binomial coefficient, and the coefficient of the third term is nine times that of the second term.  
(1) Find the values of $m$ and $n$;  
(2) If $(1+mx)^n = a_0 + a_1(x+8) + a_2(x+8)^2 + \ldots + a_n(x+8)^n$, 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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