
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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Let the function be $$f(x, y) = \left( \frac{2}{m} - \frac{m}{y} \right)^x (m > 0, y > 0)$$,
(1) ① When $m=2$, find the term with the largest binomial coefficient in the expansion of $f(4, y)$;
② If $$f(6, y) = a_{0} + \frac{a_{1}}{y} + \ldots + \frac{a_{6}}{y^{6}}$$ and $a_1 = -12$, find $$\sum\limits_{i=1}^{6}a_{i}$$;
(2) Use the binomial theorem to find the value of $$\sum\limits_{k=1}^{n}(-1)^{k}k^{2} C_{n}^{k}$$ ($n \geq 1$, $n \in \mathbb{N}^{*}$).

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