
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
The probabilities of finding a flea named Kuzi at the vertices $A, B, C,$ and $D$ of square $A B C D$ are $p_{1}, p_{2}, p_{3},$ and $p_{4}$ respectively $\left(p_{1} + p_{2} + p_{3} + p_{4} = 1\right)$. The flea jumps to the nearest vertex of the square, and it jumps equally likely to one of the two closest vertices. Find the probability of finding Kuzi at vertex $B$ after 2019 jumps.

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