
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  $a_0, a_1, \ldots, a_9$  and  $b_1 , b_2, \ldots,b_9$  be positive integers such that  $a_9<b_9$  and  $a_k \neq b_k, 1 \leq k \leq 8.$  In a cash dispenser/automated teller machine/ATM there are  $n\geq a_9$  levs (Bulgarian national currency) and for each  $1 \leq i \leq 9$  we can take  $a_i$  levs from the ATM  (if in the bank there are at least  $a_i$  levs).  Immediately after that action the bank puts  $b_i$  levs in the ATM or we take  $a_0$  levs. If we take  $a_0$  levs from the ATM the bank doesn’t put any money in the ATM. Find all possible positive integer values of  $n$  such that after finite number of takings money from the ATM there will be no money in it.

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