
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 that \( a_{i} \equiv 1 \pmod{3} \) for \( 1 \leq i \leq m \), \( b_{i} \equiv 2 \pmod{3} \) for \( 1 \leq i \leq n \), and \( c_{i} \equiv 0 \pmod{3} \) for \( 1 \leq i \leq k \), where the \( a_{i}, b_{i}, c_{i} \) are \( m+n+k \) distinct integers. If \( \sum_{i=1}^{m} a_{i}+\sum_{i=1}^{n} b_{i}+\sum_{i=1}^{k} c_{i}=2007 \), find the maximum value of \( 4m+3n+5k \).

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