
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 that $\{a\_n\}$ is an arithmetic sequence with the first term $19$ and a common difference of $-2$, and $S\_n$ is the sum of the first $n$ terms of $\{a\_n\}$.

1. Find the general term $a\_n$ and the sum $S\_n$.
2. If $\{b\_n - a\_n\}$ is a geometric sequence with the first term $1$ and a common ratio of $3$, find the general term of the sequence $\{b\_n\}$ and its sum $T\_n$.

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