
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
For the sequence $\{a_n\}$, we have $a_1 = 5$ and $a_n = 2a_{n-1} + 2^{n} - 1$ for $n \ge 2$ and $n \in \mathbb{N}^*$. If there exists a real number $\lambda$ such that the sequence $\left\{ \frac{a_n + \lambda}{2^n}\right\}$ is an arithmetic progression, then find the value of $\lambda$.

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