Let $a,$ $b,$ and $c$ be positive real numbers such that $a + b^2 + c^3 = \frac{325}{9}.$  Find the minimum value of
\[a^2 + b^3 + c^4.\]
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $a,$ $b,$ and $c$ be positive real numbers such that $a + b^2 + c^3 = \frac{325}{9}.$  Find the minimum value of
\[a^2 + b^3 + c^4.\]
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
\frac{2807}{27}
Let $a,$ $b,$ and $c$ be positive real numbers such that $a + b^2 + c^3 = \frac{325}{9}.$  Find the minimum value of
\[a^2 + b^3 + c^4.\]
train