
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
To alleviate student pressure, Class 3 of Grade 12 at a certain school held a basketball shooting game. During this event, students A and B take turns shooting baskets from a fixed position (each person shoots once per round). Under the same conditions, in each round, both students stand in the same position, and Student A shoots first. Each person shoots once, and if one of them makes a basket, that person scores 1 point while the other scores -1 point; if both make or miss the basket, both score 0 points. Assume the probability that Student A makes a basket is $\frac{2}{3}$, and the probability that Student B makes a basket is $\frac{1}{2}$, with each shot being independent of the others.
(1) After one round of shooting, let $X$ represent Student A's score. Find the probability distribution and expectation of $X$.
(2) After $n$ rounds of shooting, let $p_{i}$ denote the probability that Student A's cumulative score is lower than Student B's cumulative score after the $i^{th}$ round.
(i) Find $p_{1}, p_{2}, p_{3}$;
(ii) Given that $p_{0}=0$, it is found through computer simulation that $p_{i}=a p_{i+1}+b p_{i-1}\left(i \geqslant 1, i \in \mathbf{N}^{*}\right)$. Based on the values of $p_{1}, p_{2}, p_{3}$, determine the values of $a$ and $b$, and derive the general term formula for the sequence $\left\{p_{n}\right\}$.

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