
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
Two people, A and B, are playing a game with the following rules: a piece of paper is placed at a designated location on the ground, one of them shoots a basketball from a fixed position, if the basketball hits the paper after rebounding from the backboard, the game is successful, and the shooter continues shooting; otherwise, the game fails, and the turn switches to the other person. It is known that the probabilities of A and B shooting the ball for the first time are $\frac{2}{3}$ and $\frac{1}{3}$, respectively, and the probabilities of A and B succeeding in each game are $\frac{7}{10}$ and $\frac{4}{5}$, respectively.  
$(1)$ Find the probability that A shoots the ball for the second time.  
$(2)$ Let the probability that A shoots the ball for the $n$th time be $P_{n}$.  
$($i$)$ Express $P_{n}$ in terms of $P_{n+1}$.  
$($ii$)$ Find $P_{n}$.

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