
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
Given $a_{1}, a_{2}, \cdots, a_{n}$, there always exist 9 numbers $a_{i_{1}}, a_{i_{2}}, \cdots, a_{i_{9}}$ where $1 \leq i_{1} < i_{2} < \cdots < i_{9} \leq n$ and $b_{i} \in \{4, 7\} (i = 1, 2, \cdots, 9)$ such that $b_{1} a_{i_{1}} + b_{2} a_{i_{2}} + \cdots + b_{9} a_{i_{9}}$ is a multiple of 9.

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