
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
Two students, A and B, are playing a game. For a given real number $a_{1}$, a new real number is generated by the following method: A and B each toss a fair coin simultaneously. If both coins show heads or tails, then $a_{1}$ is multiplied by $2$ and then subtracted by $6$; if one coin shows heads and the other shows tails, then $a_{1}$ is divided by $2$ and then added by $6$. This process generates a new real number $a_{2}$, which is then operated on in the same way to obtain a new real number $a_{3}$. If $a_{3} \gt a_{1}$, A wins; otherwise, B wins. If the probability of A winning is $\frac{3}{4}$, then the range of values for $a_{1}$ is ____.

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