
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 \( a_{1}, a_{2}, \ldots, a_{2013} \) be real numbers satisfying the following conditions:
- \( a_{1}=a_{2013}=0 \)
- \( \left|a_{i}-a_{i+1}\right|<1 \), for \( 1 \leq i \leq 2012 \)
- \( \sum_{i=1}^{2013} a_{i}=0 \)

Find the greatest possible value of the sum \( \sum_{i=1}^{m} a_{i} \), where \( m \) ( \( 1 \leq m \leq 2013 \) ) is allowed to vary, in addition to the sequence \( \left\{a_{i}\right\} \).

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