
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
If a finite sequence $a_1, a_2, a_3, \ldots, a_m$ (where $m$ is a positive integer) satisfies the condition $a_1=a_m$, $a_2=a_{m-1}$, $\ldots$, $a_m=a_1$, i.e., $a_i=a_{m-i+1}$ (for $i=1, 2, \ldots, m$), we call it a "symmetric sequence". For example, the sequences $1, 2, 5, 2, 1$ and $8, 4, 2, 2, 4, 8$ are both "symmetric sequences".
1. Let $\{b_n\}$ be a 7-term "symmetric sequence", where $b_1, b_2, b_3, b_4$ form an arithmetic sequence, and $b_1=2$, $b_4=11$. List each term of $\{b_n\}$ in order;
2. Let $\{c_n\}$ be a 49-term "symmetric sequence", where $c_{25}, c_{26}, \ldots, c_{49}$ form a geometric sequence with the first term being 1 and the common ratio being 2. Find the sum $S$ of the terms of $\{c_n\}$;
3. Let $\{d_n\}$ be a 100-term "symmetric sequence", where $d_{51}, d_{52}, \ldots, d_{100}$ form an arithmetic sequence with the first term being 2 and the common difference being 3. Find the sum $S_n$ of the first $n$ terms of $\{d_n\}$ (for $n=1, 2, \ldots, 100$).

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