
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
Determine the number of  $ 8$ -tuples of nonnegative integers  $ (a_1,a_2,a_3,a_4,b_1,b_2,b_3,b_4)$  satisfying  $ 0\le a_k\le k$ , for each  $ k \equal{} 1,2,3,4$ , and  $ a_1 \plus{} a_2 \plus{} a_3 \plus{} a_4 \plus{} 2b_1 \plus{} 3b_2 \plus{} 4b_3 \plus{} 5b_4 \equal{} 19$ .

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