System of four equations of four variables including second powers. I've been tasked with solving the following system of equations and it seems like I am stuck:
$$a-x^2=y$$$$a-y^2=z$$$$a-z^2=t$$$$a-t^2=x,$$where $a$ is a real number, for which $0\leq a\leq 1$. I thought the best way would be to subtract some equations from each other and the exploit $x^2-y^2=(x+y)(x-y)$. Even some estimates could be useful, since we have an estimate for $a$. However, I put this system of equations to WolframAlfa and the solutions (depending on $a$) looked very uneasy. In general, I have got very little experience in solving quadratic systems of equations like this one, could somebody please point me in the right direction? Thanks a lot! 

Given the system,
$$a-x^2=y\\a-y^2=z\\a-z^2=t\\a-t^2=x$$
Let $(x,\,y,\,z,\,t) = (-x_1,\,-x_2,\,-x_3,\,-x_4)$ and we get,
$$x_1^2 = x_2+a\\ 
x_2^2 = x_3+a\\
x_3^2 = x_4+a\\
x_4^2 = x_1+a$$
This system was considered by Ramanujan, in particular for the case $a=5$, yielding the infinitely nested radical,
$$x_1=\small+\sqrt{5+\sqrt{5+\sqrt{5\color{red}-\sqrt{5+\sqrt{5+\sqrt{5+\sqrt{5\color{red}-\cdots}}}}}}} = \frac{2+\sqrt 5 +\sqrt{15-6\sqrt 5}}{2}=2.7472\dots$$
and similar expressions for the three other $x_i$. However it can be solved in radicals for any $a$. See this post for more details.
