Let $a,$ $b,$ $c$ be real numbers such that
\[a + 3b + 4c = a^2 + b^2 + c^2 = 25.\]Find the largest possible value of $a.$
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $a,$ $b,$ $c$ be real numbers such that
\[a + 3b + 4c = a^2 + b^2 + c^2 = 25.\]Find the largest possible value of $a.$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
\frac{25}{13}
Let $a,$ $b,$ $c$ be real numbers such that
\[a + 3b + 4c = a^2 + b^2 + c^2 = 25.\]Find the largest possible value of $a.$
train