$3.$ Consider the LSTM model in the following figure. Blue, red, and purple boxes indicate forget gate, input gate, and output gate, respectively. For each problem, explain the $(1)$ role of the state $($or gate$),(2)$ how it can be calculated, and $(3)$ the meaning of the equation.

$($a$)(4$ points$)$ Forget gate ft $($b$)(4$ points$)$ Input gate it

$($c$)(4$ points$)$ Long term state $($cell state$)$ Ct $($d$)(4$ points$)$ Output gate ot

$($e$)(4$ points$)$ Short$-$term $($hidden$)$ state ht

$($f$)(5$ points$)$ What is a potential issue for RNN$?$ How can LSTM address this issue?