Hints: Flip the Bit (Easy Version)

Read the array and the special positions; with $k = 1$, let $p$ be the (0-based) index of the unique special position after converting from $1$-based input.

On the left side, for each $i$ with $i + 1 \le p$, add $1$ to a counter when $a_i \ne a_{i + 1}$. Call this count nless.

On the right side, for each $i$ with $p \le i$ and $i + 1 < n$, add $1$ when $a_i \ne a_{i + 1}$. Call this count nmore.

Set ans to $\max(\texttt{nless}, \texttt{nmore})$. If ans is odd, increase it by $1$ so the printed value is even.

Print ans. (Per test case this is $O(n)$ time with the given loops.)

Let $x$ be the value at the special index $p$. Define $b_i = a_i \oplus x$.
Now the goal is: make all $b_i = 0$.

What does one operation do to $b$?

Answer to Hint 1:
An operation picks $[l,r]$ with $l \le p \le r$ and flips all bits there, so in array $b$ it also toggles that segment.

Because every operation contains $p$, the left side ($1..p-1$) is affected by choosing how far left we extend, and the right side ($p+1..n$) is affected by how far right we extend.

Focus on the left side first.
Set virtual values $b_0=0$ and $b_p=0$ (since $b_p=a_p\oplus x=0$).

If you scan $b_0,b_1,\dots,b_p$, every time adjacent values differ, that “boundary” must be crossed by an odd number of chosen left-extensions.

Answer to Hint 3:
Let

$$ L = \#\{\, i \in [1,p] \mid b_i \ne b_{i-1} \,\}. $$

This is the number of required odd left boundaries.

Similarly on the right, use $b_p=0$ and $b_{n+1}=0$, and define

$$ R = \#\{\, i \in [p,n] \mid b_i \ne b_{i+1} \,\}. $$

Each operation contributes exactly one left-boundary choice and one right-boundary choice.

So the number of operations must be at least both $L$ and $R$, hence at least $\max(L,R)$.

Answer to Hint 5:
It is also achievable with exactly $\max(L,R)$ operations (pair required left and right odd boundaries; extra ones can be matched on the other side in pairs).

So the minimum answer is:

$$ \boxed{\max(L,R)}. $$

Implementation is linear:

1. Read $x=a_p$.
2. Compute $L$ by scanning from $0$ to $p$ with virtual $b_0=0$, $b_p=0$.
3. Compute $R$ by scanning from $p$ to $n+1$ with virtual $b_p=0$, $b_{n+1}=0$.
4. Print $\max(L,R)$.