
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
The sequence \( 1, 2, 1, 2, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, \ldots \) is formed as follows: write down infinitely many '1's, insert a '2' between the first and the second '1's, insert two '2's between the second and the third '1's, insert three '2's between the third and the fourth '1's, and so on. If \( a_{n} \) denotes the \( n \)-th term of the sequence, find the value of \( a_{1} a_{2} + a_{2} a_{3} + \cdots + a_{2013} a_{2014} \).

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