When the polynomial $p(x)$ is divided by $x - 1,$ the remainder is 3.  When the polynomial $p(x)$ is divided by $x - 3,$ the remainder is 5.  What is the remainder when the polynomial $p(x)$ is divided by $(x - 1)(x - 3)$?
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
When the polynomial $p(x)$ is divided by $x - 1,$ the remainder is 3.  When the polynomial $p(x)$ is divided by $x - 3,$ the remainder is 5.  What is the remainder when the polynomial $p(x)$ is divided by $(x - 1)(x - 3)$?
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
x + 2
When the polynomial $p(x)$ is divided by $x - 1,$ the remainder is 3.  When the polynomial $p(x)$ is divided by $x - 3,$ the remainder is 5.  What is the remainder when the polynomial $p(x)$ is divided by $(x - 1)(x - 3)$?
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