Single Double Sum

A $6-$sided die with values $1-6$ is weighed such that the probability of rolling the values $2,3,4,5,$ and $6$ are the same whether we roll this die once or if we roll the die twice and take the sum of the two outcomes. Let $p_1$ be the probability that the value $1$ is obtained on this die. $p_1$ is the solitary positive solution to the polynomial equation

$$\displaystyle \sum_{k=1}^6 c_kp_1^k = 1$$

for some real constants $c_1,\dots, c_6$. Find $\displaystyle \sum_{k=1}^6 c_k$.