
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
Let the sum of the first $n$ terms of the sequence $\{a_n\}$ be $S_n$, and $S_n + a_n = 1$. The sequence $\{b_n\}$ is an arithmetic sequence, and $b_1 + b_2 = b_3 = 3$.

$(1)$ Find $S_n$;

$(2)$ Find the sum of the first $n$ terms of the sequence $\{a_n b_n\}$, denoted as $T_n$.

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