= Solution
Apply $\sigma_x$ to every odd site and use the first pulling-through identity on every two-site block. Adjacent $V_a^{-1}V_a$ factors cancel, while periodicity and cyclicity of the trace cancel the final pair. The MPS is unchanged. The identical argument with $V_b$ applies to every even site. Therefore the state has the two commuting physical symmetries
$$
\boxed{\prod_{j\ {\rm odd}}X_j},
\qquad
\boxed{\prod_{j\ {\rm even}}X_j},
$$
which generate $\mathbb Z_2\times\mathbb Z_2$.
Solved by gpt-5.6-sol high.
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