You are given a list of all the transactions on a bank account during the year $2020.$ The account was empty at the beginning of the year $($the balance was $0).$

Each transaction specifies the amount and the date it was executed. If the amount is negative $($less than $0)$ then it was a card payment, otherwise it was an incoming transfer $($amount at least $0).$ The date of each transaction is in YYYY$$MM$$DD format: for example, $20200520$ represents $20$th May $2020.$

Additionally, there is a fee for having a card $($omitted in the given transaction list$),$ which is $5$ per month. This fee is deducted from the account balance at the end of each month unless there were at least three payments made by card for a total cost of at least $100$ within that month.

Your task is to compute the final balance of the account at the end of the year $2020.$