
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 a finite sequence $a_1, a_2, \ldots, a_n$ (where $n$ is a positive integer) that satisfies $a_1 = a_n$, $a_2 = a_{n-1}, \ldots, a_n = a_1$, i.e., $a_i = a_{n-i+1}$ (where $i$ is a positive integer and $1 \leq i \leq n$), this sequence is called a "symmetric sequence".
1. Given that the sequence $\{b_n\}$ is a symmetric sequence with 7 terms, and $b_1, b_2, b_3, b_4$ form an arithmetic sequence, with $b_1 = 2$ and $b_4 = 11$, write down each term of $\{b_n\}$.
2. Given that $\{c_n\}$ is a symmetric sequence with $2k-1$ terms ($k \geq 1$), and $c_k, c_{k+1}, \ldots, c_{2k-1}$ form an arithmetic sequence with the first term being 50 and the common difference being -4. The sum of the first $2k-1$ terms of the sequence $\{c_n\}$ is $S_{2k-1}$. What is the value of $k$ when $S_{2k-1}$ reaches its maximum value? What is the maximum value?
3. For a given positive integer $m > 1$, write down all possible symmetric sequences with no more than $2m$ terms, such that $1, 2, 2^2, \ldots, 2^{m-1}$ are consecutive terms of the sequence; when $m > 1500$, find the sum of the first 2008 terms, $S_{2008}$, of one of these sequences.

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