
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
A computer generates even integers half of the time and another computer generates even integers a third of the time. If  $a_i$  and  $b_i$  are the integers generated by the computers, respectively, at time  $i$ , what is the probability that  $a_1b_1 +a_2b_2 +\cdots + a_kb_k$  is an even integer.

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