## Two Parallel Recurrences

Two sequences \(a(0), a(1), a(2), \ldots \) and \(b(0), b(1), b(2), \ldots \) are defined by:

- \(a(0) = b(0) = 0\),
- \(a(n+1) = a(n) + b(n) + 1\), and
- \(b(n+1) = a(n) + 2b(n) + 3\).

Part A | What is \(a(10)\)? | |
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Part B | What is the remainder when \(a(2021)\) is divided by \(10^9+7\)? | |

Part C | What is the remainder when \(a(1234567890)\) is divided by \(10^9+7\)? |