
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 a sequence $\{a_n\}$ that satisfies $a_1=1$ and $a_{n+1}=2a_n+\lambda$ (where $\lambda$ is a constant).  
(1) Investigate whether the sequence $\{a_n+\lambda\}$ is a geometric sequence and find $a_n$;  
(2) When $\lambda=1$, find the sum of the first $n$ terms, $T_n$, of the sequence $\{n(a_n+\lambda)\}$.

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