
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 the sequence ${a_n}$ with its first $n$ terms sum $S_n$, and $a_n = 2 + \left(-\frac{1}{3}\right)^{n-1}$. If for any $n \in \mathbb{N}^*$, $1 \leqslant p(S_n - 2n) \leqslant 3$, then the range of real number $p$ is \_\_\_\_\_\_.

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