
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
\( S \) is the sum of the first 7 terms of an increasing arithmetic progression \( a_{1}, a_{2}, a_{3}, \ldots \), consisting of integers. It is known that \( a_{7} a_{12} > S + 20 \) and \( a_{9} a_{10} < S + 44 \). List all possible values of \( a_{1} \).

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