
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
A finite sequence \\(\{a_n\}\\) with \\(n\\) terms is called a "symmetric sequence" if it satisfies \\(a_1 = a_n\\), \\(a_2 = a_{n-1}\\), \\(\ldots\\), \\(a_n = a_1\\), or more generally, \\(a_i = a_{n-i+1}\\) for \\(i=1,2,\ldots,n\\). For example, the sequence formed by combinations \\(C_n^0, C_n^1, \ldots, C_n^{n-1}, C_n^n\\) is a "symmetric sequence".
(i) Assume that the sequence \\(\{b_n\}\\) is a "symmetric sequence" with 7 terms, where \\(b_1, b_2, b_3, b_4\\) form a geometric progression, and \\(b_2 = 3\\), \\(b_5 = 1\\). List every term of the sequence \\(\{b_n\}\\) in order.
(ii) Let \\(\{c_n\}\\) be a "symmetric sequence" with \\(2k-1\\) terms, where \\(k∈N^*\\) and \\(k \geqslant 2\\), and satisfies \\(|c_{n+1}-c_n| = 2\\). Denote \\(S_n\\) as the sum of the first \\(n\\) terms of the sequence \\(\{c_n\}\\).
(ⅰ) If \\(c_1, c_2, \ldots, c_k\\) form a monotonically increasing sequence and \\(c_k = 2017\\), what value of \\(k\\) maximizes the value of \\(S_{2k-1}\\)?
(ⅱ) If \\(c_1 = 2018\\) and \\(S_{2k-1} = 2018\\), find the minimum value of \\(k\\).

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