
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
The sequence $\{a_n\}$ satisfies the recursion $a_n = 3a_{n-1} + 3^n - 1$ (for $n \geq 2$), and $a_1 = 5$. Find the real number $\lambda$ such that $\left\{ \frac {a_{n}+\lambda}{3^{n}} \right\}$ forms an arithmetic sequence.

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