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A mixed hypergraph is a triple \(\mathcal{H} = (X, \mathcal{C}, \mathcal{D})\), where \(X\) is the vertex set and each of \(\mathcal{C}\) and \(\mathcal{D}\) is a family of subsets of \(X\), the \(\mathcal{C}\)-edges and \(\mathcal{D}\)-edges, respectively. A proper \(k\)-coloring of \(\mathcal{H}\) is a mapping such that each \(\mathcal{C}\)-edge has two vertices with a common color and each \(\mathcal{D}\)-edge has two vertices with distinct colors. A mixed hypergraph \(\mathcal{H}\) is called circular if there exists a host cycle on the vertex set \(X\) such that every edge (\(\mathcal{C}\)- or \(\mathcal{D}\)-) induces a connected subgraph of this cycle. We propose an algorithm to color the \((3, 3)\)-uniform, complete, circular, mixed hypergraphs for every value in its feasible set. In doing so, we show: \(\chi(\mathcal{H}) = 2\) and \(\bar{\chi}(\mathcal{H}) = \frac{n}{2}\) when \(n\) is even and \(\bar{\chi}(\mathcal{H}) = \frac{n-1}{2}\) when \(n\) is odd.