
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 that $a_3 = 1$ and $S_n = a_{n+1}$ for $n = 1, 2, 3, \ldots$.
(I) Find $a_1$ and $a_2$;
(II) Find the sum of the first $n$ terms of the sequence $\{a_n\}$, denoted as $S_n$;
(III) Let $b_n = \log_2 S_n$, and suppose there exists a sequence $\{c_n\}$ such that $c_n \cdot b_{n+3} \cdot b_{n+4} = 1$. Determine the sum of the first $n$ terms of the sequence $\{c_n\}$.

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