
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
In July, a new type of clothing was sold in a market. On July 1st, only 3 pieces were sold, 6 pieces on July 2nd, 9 pieces on July 3rd, and 12 pieces on July 4th. Thereafter, the number of pieces sold each day increased by 3 until the daily sales reached a maximum for one day. After reaching the peak, the number of pieces sold began to decrease by 2 each day until exactly 3 pieces were sold on July 31st.

(1) On which day in July were the maximum number of pieces sold and what was the maximum value?

(2) According to the pattern, when the sales of this clothing reached 200 pieces, it became popular in society. But when the daily sales continuously declined and fell below 20 pieces, it was no longer popular. How many days was the clothing popular? Please explain the reason.

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