So many ^

Number Theory Level pending

What is the GCD of 2^2^2^2^2^2^2^2^2+1 (2 raised to the power of 2 8 times plus 1) and 2^2^3^4^5^6^7^8^9+1 (2 raised to the power of 2 raised to the power of 3 raised to the power of 4... raised to the power of 9 plus 1)

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