
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
There are two persons  $A,\ B$ . Given a coin of equally probability of showing up heads or tails. First  $A$  owns the coin.
Repeat the operation as follows.

(i) When  $A$  owns the coin, in tossing the coin, if the heads up, then 1 point is given to  $A$ , as the result  the coin is owend by  $A$ .

If the tails up, 0 point are given to  $A,\ B$ , as the result,  $A$  will give the coin to  $B$ .

(ii) When  $B$  owns the coin, in tossing the coin, if the heads up, then 1 point is given to  $B$ , the coin is owend by  $B$ , 

if the tails up, 0 point are given to  $A,\ B$ , as the result , $B$  will give the coin to  $A$ .

and at the time when either  $A$  or  $B$  gets 2 points, the person who gets 2 points will be a winner. 

For example, if the coin showed up like  $H,\ T,\ H,\ H$ , at the time,  $A$  gets 1 point,  $B$  gets 2 points, so  $B$  is a winner.

(1) Find the probability  $p(n)$  such that  $A$  will be a winner after exactly  $n$  tossing the coin by both of  $A$  and  $B$ .

and the following question is added to applicants for Science.

(2) Find  $\sum_{n=1}^\infty p(n)$ .

Present the answer in LaTex format: \boxed{Your answer}
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