
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 domino has two numbers (which may be equal) between 0 and 6, one at each end. The domino may be turned around. There is one domino of each type, so 28 in all. We want to form a chain in the usual way, so that adjacent dominos have the same number at the adjacent ends. Dominos can be added to the chain at either end. We want to form the chain so that after each domino has been added the total of all the numbers is odd. For example, we could place first the domino (3,4), total 3 + 4 = 7. Then (1,3), total 1 + 3 + 3 + 4 = 11, then (4,4), total 11 + 4 + 4 = 19. What is the largest number of dominos that can be placed in this way? How many maximum-length chains are there?

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