Isn't $100000$ such a huge number (Part-9)?

AG and SG both study in the same class. They are mad in love of Katrina , one of their prettiest classmates. Katrina also is fond of their intelligence and smartness.

So, on the upcoming Valentine's Day, they plan to have a date with Katrina. They fight with each other to go with Katrina on a date. Katrina listens to their fight and tries to calm down the matter.

Katrina: Hey, let's not fight over it!

SG: But I want to go with you on a date.

AG: I also want to go.

Katrina: I have an idea. Let's play a game. I have a pile of $100000$ coins and I will give you any number of coins ranging from $1$ to $100000$ . You can take out $1$ coin, $2$ coins or half of the pile of coins. You have to take turns to take out the coins. The one who takes out the last coin wins the game. Since I am a huge fan of your intelligence, I will go with the one who wins the game. SG will make the first move. So, are you ready guys?

AG: Sounds interesting!

SG: Let's play AG !!

Katrina will randomly choose how many coins to give to AG and SG . If both the players play optimally and intelligently, let the probability of AG to go on a date with Katrina is $\dfrac{a}{b}$ where $a$ and $b$ are coprime positive integers. Find $a+b$ .

Details and Assumptions:

• If the number of coins chosen by Katrina is odd, then the players can't take out half of the pile of coins. They can take out only $1$ or $2$ coins.

• Bonus points for identifying AG and SG .

• $\color{#D61F06}{\huge{Happy}}$ $\color{#D61F06}{\huge{Valentine's}}$ $\color{#D61F06}{\huge{Day}}$ .

Image credit: Cartoon released by Archie Comic Publications Inc. in 2002.