
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 \( a_{1}, a_{2}, \ldots, a_{15} \) be the days of the month when there were sunny days. Calculate the total number of drops of water that Andrei Stepanovich will drink given that for the days from the start of the month up to \( a_{1} \) he will drink nothing, from \( a_{1} \) to \( a_{2} \) (inclusive of \( a_{1} \) but exclusive of \( a_{2} \)) he will drink one drop per day, and so on. We need to find the final expression for the total drops drunk.

Additionally, determine the number of drops that Ivan Petrovich drinks, which is equal to the sum of the days not included in \( a_{i} \).

The expressions are given by:
\[ 1 \cdot (a_{2} - a_{1}) + 2 \cdot (a_{3} - a_{2}) + \ldots + 14 \cdot (a_{15} - a_{14}) + 15 \cdot (30 - a_{15} + 1) = 15 \cdot 31 - (a_{1} + a_{2} + \ldots + a_{15}) \]

For Ivan Petrovich:
\[ (1 + 2 + \ldots + 30) - (a_{1} + a_{2} + \ldots + a_{15}) = 15 \cdot 31 - (a_{1} + a_{2} + \ldots + a_{15}) \]

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