
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 $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, and $S_n=2a_n-1$ ($n\in\mathbb{N}^*$).

(1) Find the general formula for the sequence $\{a_n\}$.

(2) If the sequence $\{b_n\}$ satisfies $b_n=2n\cdot a_n$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n$.

(3) If the sequence $\{c_n\}$ satisfies $c_n=3^n+2(-1)^{n-1}\lambda a_n$ (where $\lambda$ is a non-zero constant), determine the range of values for $\lambda$ such that $c_{n+1} > c_n$ for all $n\in\mathbb{N}^*$.

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