
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
Let the initial price of the product be represented by $P$. The price of the product increased by 10% in January, decreased by 15% in February, increased by 20% in March, decreased by 10% in April, and increased by y% in May. The price after each period is represented as follows:
1. $P + 0.10P = 1.10P$
2. $1.10P - 0.15(1.10P) = 0.95(1.10P) = 1.045P$
3. $1.045P + 0.20(1.045P) = 1.247P$
4. $1.247P - 0.10(1.247P) = 1.1237P$
5. $1.1237P + y\% \text{ of }1.1237P = 1.1237P(1 + \frac{y}{100})$
The price of the product at the end of May was the same as it was at the beginning of January. Therefore, the equation is:
1.1237P(1 + \frac{y}{100}) = P

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