Hints: Exceptional Segments

Hint 1 ↕

Turn the segment XOR condition into a prefix-XOR condition.

If $p_i = 1 \oplus 2 \oplus \cdots \oplus i$ (and $p_0=0$), then

$$ l \oplus (l+1)\oplus\cdots\oplus r = p_r \oplus p_{l-1}. $$

So we need $p_r = p_{l-1}$ with $l \le x \le r$.

Hint 2 ↕

Let $a=l-1$ and $b=r$.

Then $a \in [0, x-1]$, $b \in [x, n]$, and we count pairs $(a,b)$ such that $p_a=p_b$. So this is a frequency-matching problem between two disjoint index ranges.

Hint 3 ↕

Use the classic pattern for XOR of first integers:

$$ p_i = \begin{cases} i, & i \bmod 4 = 0\\ 1, & i \bmod 4 = 1\\ i+1, & i \bmod 4 = 2\\ 0, & i \bmod 4 = 3 \end{cases} $$

So $p_i$ only depends on $i \bmod 4$.

Hint 4 ↕

For values other than $0$ and $1$, each value appears at only one index:

if value $\equiv 0 \pmod 4$, it comes from exactly one $i \equiv 0 \pmod 4$;
if value $\equiv 3 \pmod 4$, it comes from exactly one $i \equiv 2 \pmod 4$.

Since ranges $[0,x-1]$ and $[x,n]$ are disjoint, those unique values cannot match across ranges.

Hint 5 ↕

So only repeated values matter:

$p_i=0$ for $i=0$ and all $i \equiv 3 \pmod 4$;
$p_i=1$ for all $i \equiv 1 \pmod 4$.

If

$$ z_L=\#\{i\in[0,x-1]:p_i=0\},\quad z_R=\#\{i\in[x,n]:p_i=0\}, $$ $$ o_L=\#\{i\in[0,x-1]:p_i=1\},\quad o_R=\#\{i\in[x,n]:p_i=1\}, $$

then

$$ \text{answer}=z_L z_R + o_L o_R. $$

Hint 6 ↕

Count residues in a range in $O(1)$:

$$ \#\{i\in[L,R]: i \bmod 4 = r\} = \left\lfloor\frac{R-r}{4}\right\rfloor - \left\lfloor\frac{L-1-r}{4}\right\rfloor $$

(with the usual $0$ handling when $L>R$).

Then each test case is $O(1)$, total $O(t)$.