Let $a = e^{4 \pi i/5}$ and $b = e^{2 \pi i/17}.$  Compute the value of the product
\[(a + b)(a + b^{16})(a^2 + b^2)(a^2 + b^{15})(a^3 + b^8)(a^3 + b^9)(a^4 + b^4)(a^4 + b^{13}).\]
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $a = e^{4 \pi i/5}$ and $b = e^{2 \pi i/17}.$  Compute the value of the product
\[(a + b)(a + b^{16})(a^2 + b^2)(a^2 + b^{15})(a^3 + b^8)(a^3 + b^9)(a^4 + b^4)(a^4 + b^{13}).\]
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
1
Let $a = e^{4 \pi i/5}$ and $b = e^{2 \pi i/17}.$  Compute the value of the product
\[(a + b)(a + b^{16})(a^2 + b^2)(a^2 + b^{15})(a^3 + b^8)(a^3 + b^9)(a^4 + b^4)(a^4 + b^{13}).\]
train