
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
Assume that \( a_{i} \in \{1, -1\} \) for all \( i=1, 2, \ldots, 2013 \). Find the least positive value of the following expression:
\[ 
\sum_{1 \leq i < j \leq 2013} a_{i} a_{j} 
\]

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