
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 the constant $a \neq 0$, the sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n$, with $a_1 = 1$ and $$a_{n}= \frac {S_{n}}{n}+a(n-1).$$
Find the general term formula for the sequence $\{a_n\}$ and the range of values for the real number $a$ if the sequence $\{b_n\}$ is strictly increasing, where $$b_{n}=3^{n}+(-1)^{n}a_{n}.$$

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