Prove/Disprove: When $A,B,C$ are symmetric matrices, the transpose of $ABC$ is $CBA$

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Prove/Disprove the following: When $A,B,C$ are symmetric matrices, the transpose of $ABC$ is $CBA$

This was what I did:

$$\text{Symmetric} \ A,B,C \implies (A^T = A) \land (B^T = B) \land (C^T = C) \ \ \ \ \ \ \ (1)$$ $$(ABC)^T = C^TB^TA^T = CBA \ \ \ \ \ \ \text{(By (1))} $$

But according to the solutions manual $$(ABC)^T = C^TB^TA^T \neq CBA$$ Why would that be so?

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