
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 $\left\{ a_n \right\}$ is a geometric sequence, and the sum of the first $n$ terms is $S_n$, and $S_n = 2^n + a \left( n\in N^* \right)$.

(1) Find the value of $a$ and the general formula for the sequence $\left\{ a_n \right\}$;

(2) If $b_n = n a_n$, find the sum of the first $n$ terms $T_n$ of the sequence $\left\{ b_n \right\}$.

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