Counterexample:If the second-order Taylor polynomial of f at $\textbf{a}$ is f, then the third-order Taylor polynomial of f at $\textbf{a}$ is also f.

Question

I encountered this question about the multivariable Taylor polynomial.

True or false: Let $f:\mathbb{R}^2\rightarrow\mathbb{R}$ be a function of class $C^3$, and let $\textbf{a}\in \mathbb{R}^2$. If the second-order Taylor polynomial of f at $\textbf{a}$ is f, then the third-order Taylor polynomial of f at $\textbf{a}$ is also f.

Intuitively, I think this is false, but I cannot find a counterexample satisfying all the conditions, especially to satisfy the second-order Taylor polynomial of f at $\textbf{a}$ is f.

Can anyone help me? Thanks in advance.