
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
José wants to fill the cells of a $2 \times n$ board with zeros and ones such that two identical neighboring numbers in the same row prevent the corresponding cells of the other row from being filled with identical numbers. For example, in the following configuration, the values of $A$ and $B$ cannot be the same.

| 0 | 1 | 0 | $\cdots$ | 1 | 1 | $\cdots$ |
| :--- | :--- | :--- | :--- | :--- | :--- | :--- |
| 1 | 1 | 0 | $\cdots$ | $A$ | $B$ | $\cdots$ |

a) Find all possible fillings of the following board:

| 0 | 0 | 1 |
| :--- | :--- | :--- |
| 1 |  |  |

b) On the board, all columns are filled except for the last two. In how many ways can the numbers in cells $A$ and $B$ be chosen?

| $\cdots$ | 0 | $A$ |
| :--- | :--- | :--- |
| $\cdots$ | 1 | $B$ |

c) In how many ways can José fill the board if $n=2020$?



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