Suppose you have a two$-$layer neural network with $3$ hidden neurons, and $1$ output neuron. Each hidden neuron accepts $3-$dimensional input, is tanh$-$activated with the following form:

hi$($x$)=$tanh$($w$$ix$)$ where $1$i$3.$

Next, the output neuron is o$($z$)=$$$z where z$=[$z$1,$z$2,$z$3]$ with zi being the output of the corresponding hidden neuron hi$($x$)$

Let us initialize wi$=[0\mathrm{.}1,0\mathrm{.}1,0\mathrm{.}1]$ for all hidden neuron. Let's also initialize $=[0\mathrm{.}0,0\mathrm{.}1,0\mathrm{.}2].$ Note that these notations suggest that wi and are treated as column vectors, i$.$e$.,[0\mathrm{.}1,0\mathrm{.}1,0\mathrm{.}1]$ is a row vector so $[0\mathrm{.}1,0\mathrm{.}1,0\mathrm{.}1]$ is a column vector.

$(1)$ Calculate the output of the network for the input x$=[0,1,1].(10$pts$)$

$(2)$ Update all of the weights using backpropagation given that the desired$/$target output is $2\mathrm{.}0.$ Use a learning rate of $0\mathrm{.}5.(10$pts$)$