To solve the problem, we analyze the Fibonacci sequence modulo 3 and identify the pattern of terms removed (divisible by 3) and the resulting sequence.

Step 1: Fibonacci Sequence Modulo 3

The Fibonacci sequence modulo 3 has a repeating cycle of length 8:
1, 1, 2, 0, 2, 2, 1, 0

Terms divisible by 3 are at positions 4, 8, 12, ... (multiples of 4).

Step 2: Kept Terms Pattern

For each cycle of 8 original terms, 6 terms are kept (excluding positions 4 and 8). The mod3 values of these kept terms form a periodic sequence of length 6:
1, 1, 2, 2, 2, 1

Step 3: Find the 2023rd Kept Term

The 2023rd term in the kept sequence corresponds to the remainder when 2023 is divided by 6:
2023 = 6×337 + 1

The remainder is 1, so the 2023rd term is the first element of the periodic sequence of kept terms (mod3).

Answer: 1

$\boxed{1}$

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