3. Let real numbers $a$, $b$, $c$ satisfy the polynomial
$$
P(x)=x^{4}+a x^{3}+b x^{2}+a x+c
$$

has exactly three distinct real roots: $\tan y$, $\tan 2y$, $\tan 3y$ $(y \in \mathbf{R})$. Find all $y \in [0, \pi)$.
3. Let the real roots of the polynomial $P(x)$ be $r, r, s, t$.
$$
\begin{array}{l}
\text { By } P(x)=(x-r)^{2}(x-s)(x-t) \\
=x^{4}+a x^{3}+b x^{2}+a x+c \\
\Rightarrow 2 r+s+t=r^{2} s+r^{2} t+2 r s t \\
\Rightarrow 2 r(1-s t)=\left(r^{2}-1\right)(s+t) .
\end{array}
$$

If $r^{2}-1=0$, then $1-s t=0$. Conversely, if $1-s t=0$, then $r^{2}-1=0$.
Assume $r^{2}=s t=1, 0 \leqslant y<\pi$, then there are the following three possibilities.
(1) If $r=\tan y$, then $y \in\left\{\frac{\pi}{4}, \frac{3 \pi}{4}\right\}$.
Thus, $\tan 2 y$ is undefined, which is a contradiction.
(2) If $r=\tan 2 y$, then
$$
y \in X \triangleq\left\{\frac{\pi}{8}, \frac{3 \pi}{8}, \frac{5 \pi}{8}, \frac{7 \pi}{8}\right\} .
$$

Let $s=\tan y, t=\tan 3 y$. Then, for each $y \in X, y+3 y=4 y$ is an odd multiple of $\frac{\pi}{2}$.
Thus, $st = \tan y \cdot \tan 3 y=1$.
(3) If $r=\tan 3 y$, then
$$
st = \tan y \cdot \tan 2 y=1.
$$

Thus, $\tan 3 y$ is undefined, which is a contradiction.
If $r^{2}-1 \neq 0$ and $1-s t \neq 0$, divide both sides of the equation
$$
2 r(1-s t)=\left(r^{2}-1\right)(s+t)
$$

by $\left(r^{2}-1\right)(1-s t)$ to get
$$
\frac{2 r}{1-r^{2}}+\frac{s+t}{1-s t}=0.
$$

We discuss the following three cases.
(i) If $r=\tan y, s=\tan 2 y, t=\tan 3 y$, then by equation (1) we get $\tan 2 y+\tan 5 y=0$.
Thus, $y \in\left\{\frac{\pi}{7}, \frac{2 \pi}{7}, \cdots, \frac{6 \pi}{7}\right\}$.
(ii) If $r=\tan 2 y, s=\tan y, t=\tan 3 y$, then by equation (1) we get $\tan 4 y+\tan 4 y=0$.

When $y=\frac{\pi}{2}, \frac{\pi}{4}, \frac{3 \pi}{4}$, $\tan y$ or $\tan 2 y$ is undefined.
Thus, $y \in\left\{\frac{\pi}{8}, \frac{3 \pi}{8}, \frac{5 \pi}{8}, \frac{7 \pi}{8}\right\}$.
And when $y \in\left\{\frac{\pi}{8}, \frac{3 \pi}{8}, \frac{5 \pi}{8}, \frac{7 \pi}{8}\right\}$,
$$
r=\tan 2 y=1 \text { or }-1.
$$

This is a contradiction.
(iii) If $r=\tan 3 y, s=\tan y, t=\tan 2 y$, then by equation (1) we get $\tan 6 y+\tan 3 y=0$.
Thus, $y \in\left\{\frac{\pi}{9}, \frac{2 \pi}{9}, \frac{\pi}{3}, \frac{4 \pi}{9}, \frac{5 \pi}{9}, \frac{2 \pi}{3}, \frac{7 \pi}{9}, \frac{8 \pi}{9}\right\}$.
In summary, the values of $y$ that satisfy the conditions are
$$
\begin{array}{r}
y \in\left\{\frac{\pi}{8}, \frac{3 \pi}{8}, \frac{5 \pi}{8}, \frac{7 \pi}{8}, \frac{\pi}{7}, \frac{2 \pi}{7}, \frac{3 \pi}{7}, \frac{4 \pi}{7}, \frac{5 \pi}{7},\right. \\
\left.\frac{6 \pi}{7}, \frac{\pi}{9}, \frac{2 \pi}{9}, \frac{\pi}{3}, \frac{4 \pi}{9}, \frac{5 \pi}{9}, \frac{2 \pi}{3}, \frac{7 \pi}{9}, \frac{8 \pi}{9}\right\} .
\end{array}
$$