
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 an arithmetic sequence $\{a_n\}$ where $S_n$ denotes the sum of the first $n$ terms, and it satisfies $a_2 + a_4 = 6$ and $a_6 = S_3$.
(Ⅰ) Find the general term formula for $\{a_n\}$.
(Ⅱ) If $k \in \mathbb{N}^*$ and $a_k, a_{3k}, S_{2k}$ form a geometric sequence, find the value of $k$.

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