You are tasked to design a web crawler that explores a website's structure. The website is represented as a graph, where each web page is a node, and each hyperlink between web pages is an edge. The crawler starts from a given page and follows hyperlinks, exploring as deeply as possible before backtracking to other links.

Consider the following website structure:

Each edge represents a hyperlink between two web pages.

Constraints:

$-$ The crawler must visit each web page $($node$)$ only once.

$-$ The crawler should use a DFS approach to explore the site.

Tasks:

$1.$ Write a Depth$-$First Search $($DFS$)$ algorithm to explore this website's graph structure, starting from page $\backslash (\backslash$mathbf$\{$A$\}\backslash ).$

$2.$ The crawler should visit each page $($node$)$ and explore all the hyperlinks $($edges$)$ starting from A$.$

$3.$ Provide the path followed by the DFS crawler, starting from $\backslash (\backslash$mathbf$\{$A$\}\backslash )$ and visiting all the reachable pages.

Requirements:

$1.$ Formulate an efficient algorithm to perform the above task using DFS$.$

$2.$ Provide a description about the design used and the rationale behind the design choice.

$3.$ Analyze the time complexity of the algorithm.

$4.$ Implement the above problem statement using Python $3\mathrm{.}7$ and above.