
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
A machine accepts coins of  $k{}$  values  $1 = a_1 <\cdots < a_k$  and sells  $k{}$  different drinks with prices  $0<b_1 < \cdots < b_k$ . It is known that if we start inserting coins into the machine in an arbitrary way, sooner or later the total value of the coins will be equal to the price of a drink. For which sets of numbers  $(a_1,\ldots,a_k;b_1,\ldots,b_k)$  does this property hold?

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