
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 arithmetic sequence $\{a_n\}$ has the first term $a$ and common difference $b$, and the geometric sequence $\{b_n\}$ has the first term $b$ and common ratio $a$.
(Ⅰ) If the sum of the first $n$ terms of the sequence $\{a_n\}$ is $S_n = -n^2 + 3n$, find the values of $a$ and $b$.
(Ⅱ) If $a \in \mathbb{N}^{+}$ and $b \in \mathbb{N}^{+}$, and $a < b < a_2 < b_2 < a_3$,
(i) find the value of $a$;
(ii) for the sequences $\{a_n\}$ and $\{b_n\}$, given the relationship $a_n + k = b_n$ where $k$ is a constant and $k \in \mathbb{N}^{+}$, find the maximum value of $b$.

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