
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 \( a_{0}, a_{1}, \ldots, a_{k} \) (with \( k \geq 1 \)) be positive integers. Find all positive integers \( y \) such that
\[ a_{0} \mid y, \ (a_{0}+a_{1}) \mid (y+a_{1}), \ \ldots, \ (a_{0}+a_{k}) \mid (y+a_{k}). \

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