
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 the binomial $(x+3x^{2})^{n}$.

$(1)$ If the sum of the binomial coefficients is $128$.

    $(i)$ Find the term with the largest binomial coefficient in the expansion;

    $(ii)$ Find the term with the largest coefficient in the expansion;

$(2)$ If $x=3$ and $n=2016$, find the remainder when the value of the binomial is divided by $7$.

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