
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
A teacher and students in a certain class are asked questions. The probability that the teacher's answer will be correct is $\alpha$, and the probability of a correct answer from a student is either $\beta$ or $\gamma$ depending on whether the student is a boy or a girl, respectively. The probability that the answer of a randomly chosen student matches the teacher’s answer is $1 / 2$. Find the ratio of the number of boys to the number of girls in the class.

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