
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 $S_n$ be the sum of the first $n$ terms of the sequence $\{ a_n \}$, given that $a_1 \neq 0$, $2a_n - a_1 = S_1 S_n$, for all $n \in \mathbb{N}^*$.
(I) Find $a_1$, $a_2$, and the general term formula for the sequence $\{ a_n \}$;
(II) Find the sum of the first $n$ terms of the sequence $\{ n a_n \}$.

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