
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
Players A and B are playing a game involving a given real number $a_1$. To generate a new real number, they perform one of the following operations depending on the outcome of simultaneously tossing two fair coins: If they both show heads or both show tails, they multiply $a_1$ by 2 and then subtract 12; if one coin shows heads and the other tails, they divide $a_1$ by 2 and then add 12. This results in a new real number $a_2$. Applying the same procedure to $a_2$ results in $a_3$. If $a_3 > a_1$, then player A wins; otherwise, player B wins. If the probability of player A winning is $\frac{3}{4}$, find the range of possible values for $a_1$.

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