
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
Consider a  $2 \times n$  grid where each cell is either black or white, which we attempt to tile with  $2 \times 1$  black or white tiles such that tiles have to match the colors of the cells they cover. We first randomly select a random positive integer  $N$  where  $N$  takes the value  $n$  with probability  $\frac{1}{2^n}$ . We then take a  $2 \times N$  grid and randomly color each cell black or white independently with equal probability. Compute the probability the resulting grid has a valid tiling.

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