In a data structure, there is a doubly linked list with 'head' and 'tail'

pointers for storing $n$ elements. When searching for a specific element,

the search starts from the head and proceeds until the target element is

found, at an index i where $in.$ The cost for each 'search' operation

is $i,$ and when the target element is found, it's moved towards the head

of the list by $1$ index at a cost of $c($a positive constant$).$ We want to

calculate the amortized time complexity for searching and moving an

element in the worst$-$case scenario. Assume that the number of times

the target element is searched is less than or equal to the total number

of elements in the list, and consider a scenario where we search for this

specific element repeatedly.

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