
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 that $f(i,k)=i⋅2^{(k-1)}$ where $i \in \mathbb{N}^*, k \in \mathbb{N}^*$, and for example, $f(2,3)=2×2^{(3-1)}=8$. For positive integers $m$ and $n$, when $m \geqslant 2$ and $n \geqslant 2$, let $g(i,n)=f(2^{0},n)+f(2^{1},n)+…+f(2^{i},n)$ and $S(m,n)= \sum\limits\_{i=1}^{m}(-1)^{i}g(i,n)$. Find $S(4,6)=$ _______ .

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