Assume that $X=\{x^{(1)},...,x^{(m)}\}$ is a dataset of $m$ samples with $3$ features. The samples are classified into $2$ categories with labels $y^{(i)}in\{0,1\}.$ We want to perform binary classification using a simple neural network. Assume that the three features are $x_1,x_2$ and $x_3,$the three neurons in the hidden layer are $h_1,h_2$ and $h_3,$ the output neuron is $o,$ the bias for input layer is $b_x$ and that for the hidden layer is $b_h.$ Let the weight from $x_i$ to $h_j$ be $w_{i,j}^{[1]}$ for $iin\{1,2,3\},jin\{1,2,3\},$ and the weight from $h_j$ to $o$ be $w_j^{[2]}.$ Suppose we use the sigmoid function

as the activation function for $h_1,h_2,h_3$ and $o,$ the loss function: $l=\frac{1}{m}{\displaystyle \underset{i=1}{\overset{m}{}}}(o^{(i)}-y^{(i)}{)}^2,$

where $o^{(i)}$ is the result of the output neuron for example $i.$

The figure: The illustration of three connected neural networks consists of INPUT, HIDDEN and OUTPUT layers, as well as the related weight and bias.

$($a$)$ First, please distinguish which neural network is correct and suitable for the binary classification task from the above figure and derive the final output for one training sample using forward propagation based on the given architecture. Your answer should be written in terms of $x,w,b$ and the sigmoid function.

$($b$)($Based on the previous background$)$ Second, for the correct neural network, if $we$ use a leaming rate of $,$ please derive the updating formula for $w_1^{[2]}.$ Your answer shoule be written in terms of $o^{(i)},y^{(i)}($if you use some additional symbols, please add some note for them$).$