
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 positive sequence $\{a_{n}\}$ is a geometric sequence with a common ratio of $\frac{1}{2}$, and the general formula of sequence $\{b_{n}\}$ is ${b}_{n}=\frac{n}{2}$. If there are exactly $3$ positive integers $n$ such that $a_{n} \gt b_{n}$, then the range of values for $a_{1}$ is ______.

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