
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 $a_{1} \neq 0$, and $2a_{n} - a_{1} = S_{1}S_{n}$ for all $n \in \mathbb{N}^*$. Find:

1. The value of $a_{1}$ and the general term formula for the sequence ${a_{n}}$.
2. The sum of the first $n$ terms, $T_{n}$, of the sequence ${na_{n}}$.

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