
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
In a sequence of natural numbers  $ a_1 $ ,  $ a_2 $ ,  $ \dots $ ,  $ a_ {1999} $ ,  $ a_n-a_ {n-1} -a_ {n-2} $  is divisible by  $ 100 (3 \leq n \leq 1999) $ . It is known that  $ a_1 =  19$  and  $ a_2 =  99$ . Find the remainder of  $ a_1 ^ 2 + a_2 ^ 2 + \dots + a_ {1999} ^ 2 $  by  $8$ .

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