
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 $(a\_1)$, $(a\_2)$, $(a\_3)$, $(…)$, $(a\_m (m$ is a positive integer$))$ that satisfies the conditions $(a\_1=a\_m)$, $(a\_2=a_{m-1})$, $(…)$, $(a\_m=a\_1)$, i.e., $(a\_i=a_{m-i+1} (i=1,2,…,m))$, we call it a "symmetrical sequence". For example, the sequences $1$, $2$, $5$, $2$, $1$ and $8$, $4$, $2$, $2$, $4$, $8$ are both "symmetrical sequences".

1. Let $(b\_n)$ be a "symmetrical sequence" of 7 items, where $(b\_1)$, $(b\_2)$, $(b\_3)$, $(b\_4)$ form an arithmetic sequence, and $(b\_1=2)$, $(b\_4=11)$. Write out each item of $(b\_n)$ in order.
2. Let $(c\_n)$ be a "symmetrical sequence" of 49 items, where $(c_{25})$, $(c_{26})$, $(…)$, $(c_{49})$ form a geometric sequence with the first term being $1$ and the common ratio being $2$. Find the sum $S$ of all items in $(c\_n)$.

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