
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 particle located at point $A$ (Fig. 62) has a probability $p_{11}$ of staying at $A$ after a unit of time and a probability $p_{12}$ of moving to $B$. If it is at $B$, it has a probability $p_{22}$ of staying at $B$ after a unit of time and a probability $p_{21}$ of moving to $A$. What is the probability that after $n$ steps (units of time) the particle will be at point $A$ if it initially starts at:

a) point $A$? 

b) point $B$?

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