Let $z = a + bi,$ where $a$ and $b$ are positive real numbers.  If
\[z^3 + |z|^2 + z = 0,\]then enter the ordered pair $(a,b).$
<|im_start|>system
You are a helpful assistant.<|im_end|>
<|im_start|>user
Let $z = a + bi,$ where $a$ and $b$ are positive real numbers.  If
\[z^3 + |z|^2 + z = 0,\]then enter the ordered pair $(a,b).$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
\left( \frac{1}{2}, \frac{\sqrt{7}}{2} \right)
Let $z = a + bi,$ where $a$ and $b$ are positive real numbers.  If
\[z^3 + |z|^2 + z = 0,\]then enter the ordered pair $(a,b).$
train