Assume vectors $\boldsymbol{a}_1, \boldsymbol{a}_2, \ldots, \boldsymbol{a}_m$ are linearly independent in a vector space $V$, where $m$ is an integer greater than or equal to 3. Obtain the condition that $m$ must satisfy in order for $\boldsymbol{a}_1 + \boldsymbol{a}_2,\ \boldsymbol{a}_2 + \boldsymbol{a}_3,\ \ldots,\ \boldsymbol{a}_{m-1} + \boldsymbol{a}_m$ and $\boldsymbol{a}_m + \boldsymbol{a}_1$ to be linearly independent.