
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
If for any positive integer $n$, there always exists a positive integer $m$ such that the sum of the first $n$ terms $S_n$ of the sequence $\{a_n\}$ is equal to $a_m$, then we call $\{a_n\}$ a "recursive sequence".  
(I) ① Is the sequence $\{a_n\}$ with the sum of the first $n$ terms being $S_n = 2^n$ a "recursive sequence"? Please explain your reasoning.  
② Is the sequence $\{b_n\}$ with the general term $b_n = 2n$ a "recursive sequence"? Please explain your reasoning.  
(II) Let $\{a_n\}$ be an arithmetic sequence with the first term $a_1 = 1$ and the common difference $d < 0$. If $\{a_n\}$ is a "recursive sequence", determine the value of $d$.  
(III) Is it true that for any arithmetic sequence $\{a_n\}$, there always exist two "recursive sequences" $\{b_n\}$ and $\{c_n\}$ such that the equality $a_n = b_n + c_n$ (for $n \in \mathbb{N^*}$) holds? Please provide your conclusion and explain your reasoning.

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