
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  $(b_n)_{n \ge 0}=\sum_{k=0}^{n} (a_0+kd)$  for positive integers  $a_0$  and  $d$ . We consider all such sequences containing an element  $b_i$  which equals  $2010$ . Determine the greatest possible value of  $i$  and for this value the integers  $a_0$  and  $d$ .

*(41th Austrian Mathematical Olympiad, regional competition, problem 4)*

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