The forget gate and the input gate together also update the cell state with the following equation,
$$ \mathbf{c}^{(t)} = \mathbf{f}^{(t)} \otimes \mathbf{c}^{(t-1)} + \mathbf{i}^{(t)} \otimes \mathbf{\tilde{c}}^{(t)}, $$where \( f^{(t)} \) and \( i^{(t)} \) are the outputs of the forget gate and the input gate, respectively.