
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 sequence of positive terms $\{a\_n\}$ with the first term $1$, it satisfies the condition $a_{n+1}^{2}+a_{n}^{2} < \frac{5}{2}a_{n+1}a_{n}$, where $n\in\mathbb{N^*}$. If $a_{2}=\frac{3}{2}$, $a_{3}=x$, and $a_{4}=4$, find the range of values for $x$.

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