
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
Find a number  $n \geq 9$  such that for any  $n$  numbers, not necessarily distinct,  $a_1,a_2, \ldots , a_n$ , there exists 9 numbers  $a_{i_1}, a_{i_2}, \ldots , a_{i_9}, (1 \leq i_1 < i_2 < \ldots < i_9 \leq n)$  and  $b_i \in {4,7}, i =1, 2, \ldots , 9$  such that  $b_1a_{i_1} + b_2a_{i_2} + \ldots + b_9a_{i_9}$  is a multiple of 9.

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