Vector Algebra: JEE Main 2023 Mathematics question

Let $\vec{a}=2 \hat{i}+\hat{j}+\hat{k}$, and $\vec{b}$ and $\vec{c}$ be two nonzero vectors such that $|\vec{a}+\vec{b}+\vec{c}|=|\vec{a}+\vec{b}-\vec{c}|$ and $\vec{b} \cdot \vec{c}=0$. Consider the following two statements: (A) $|\vec{a}+\lambda \vec{c}| \geq|\vec{a}|$ for all $\lambda \in \mathbb{R}$. (B) $\vec{a}$ and $\vec{c}$ are always parallel. Then,