
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
Qing initially writes the ordered pair  $(1,0)$  on a blackboard. Each minute, if the pair  $(a,b)$  is on the board, she erases it and replaces it with one of the pairs  $(2a-b,a)$ ,  $(2a+b+2,a)$  or  $(a+2b+2,b)$ .  Eventually, the board reads  $(2014,k)$  for some nonnegative integer  $k$ .  How many possible values of  $k$  are there?

*Proposed by Evan Chen*

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