
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
Let $A$ be an array consisting of $n$ ordered real numbers, denoted as: $A=(a_1, a_2, \ldots, a_i, \ldots, a_n)$. Here, $a_i$ ($i=1, 2, \ldots, n$) is called the "element" of array $A$, and $S$ is referred to as the subscript of $A$. If each "element" in array $S$ comes from elements with different subscripts in array $A$, then $A=(a_1, a_2, \ldots, a_n)$ is called a subarray of $B=(b_1, b_2, \ldots, b_n)$. Define the relation number of two arrays $A=(a_1, a_2, \ldots, a_n)$ and $B=(b_1, b_2, \ldots, b_n)$ as $C(A, B)=a_1b_1+a_2b_2+\ldots+a_nb_n$.
(Ⅰ) If $A=(-\frac{1}{2}, \frac{1}{2})$, $B=(-1, 1, 2, 3)$, and $S$ is a subarray of $B$ containing two "elements", find the maximum value of $C(A, S)$;
(Ⅱ) If $A=(\frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3}, \frac{\sqrt{3}}{3})$, $B=(0, a, b, c)$, and $a^2+b^2+c^2=1$, with $S$ being a subarray of $B$ containing three "elements", find the maximum value of $C(A, S)$.

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