
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
All the positive integers which are co-prime to 2012 are grouped in increasing order in such a way that the \( n^{\text{th}} \) group has \( 2n - 1 \) numbers. For example, the first three groups are (1), (3, 5, 7), (9, 11, 13, 15, 17). It is known that 2013 belongs to the \( k^{\text{th}} \) group. Find the value of \( k \). (Note: Two integers are said to be co-prime if their greatest common divisor is 1.)

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