
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 15 terms of an increasing arithmetic progression $a_{1}, a_{2}, a_{3}, \ldots$ consisting of integers. It is known that $a_{7} a_{16}>S-24$ and $a_{11} a_{12}<S+4$. Determine all possible values of $a_{1}$.

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