
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
How many non-decreasing tuples of integers  $(a_1, a_2, \dots, a_{16})$  are there such that  $0 \leq a_i \leq 16$  for all  $i$ , and the sum of all  $a_i$  is even?

*Proposed by Nancy Kuang*

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