
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
A Markov chain is named after the Russian mathematician Andrey Markov, and its process is characterized by the "memoryless" property, meaning that the probability distribution of the $n+1$-th state only depends on the $n$-th state and is independent of the states $n-1, n-2, n-3, \ldots$. 

Consider two boxes, Box A and Box B, each containing 2 red balls and 1 black ball of the same size, shape, and texture. One ball is randomly selected from each of the two boxes and exchanged. This operation is repeated $n$ times ($n \in \mathbf{N}^{*}$). Let $X_n$ denote the number of black balls in Box A after $n$ operations. Let $a_n$ be the probability that there is exactly 1 black ball in Box A, and let $b_n$ be the probability that there are exactly 2 black balls in Box A.

(1) Find the distribution of $X_{1}$;
(2) Find the general term formula for the sequence $\left\{a_{n}\right\}$;
(3) Find the mathematical expectation of $X_{n}$.

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