
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
Three strictly increasing sequences
\[a_1, a_2, a_3, \ldots,\qquad b_1, b_2, b_3, \ldots,\qquad c_1, c_2, c_3, \ldots\]
of positive integers are given. Every positive integer belongs to exactly one of the three sequences. For every positive integer  $n$ , the following conditions hold:
(a)  $c_{a_n}=b_n+1$ ;
(b)  $a_{n+1}>b_n$ ;
(c) the number  $c_{n+1}c_{n}-(n+1)c_{n+1}-nc_n$  is even.
Find  $a_{2010}$ ,  $b_{2010}$  and  $c_{2010}$ .

*(4th Middle European Mathematical Olympiad, Team Competition, Problem 1)*

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