
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 a positive sequence $\{a_n\}$ with the first term being 1, it satisfies $a_{n+1}^2 + a_n^2 < \frac{5}{2}a_{n+1}a_n$, where $n \in \mathbb{N}^*$, and $S_n$ is the sum of the first $n$ terms of the sequence $\{a_n\}$.
1. If $a_2 = \frac{3}{2}$, $a_3 = x$, and $a_4 = 4$, find the range of $x$.
2. Suppose the sequence $\{a_n\}$ is a geometric sequence with a common ratio of $q$. If $\frac{1}{2}S_n < S_{n+1} < 2S_n$ for $n \in \mathbb{N}^*$, find the range of $q$.
3. If $a_1, a_2, \ldots, a_k$ ($k \geq 3$) form an arithmetic sequence, and $a_1 + a_2 + \ldots + a_k = 120$, find the minimum value of the positive integer $k$, and the corresponding sequence $a_1, a_2, \ldots, a_k$ when $k$ takes the minimum value.

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