
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 \( M \) starting from the origin moves according to the vector \( \boldsymbol{a} = (0,1) \) with a probability of \( \frac{2}{3} \). It moves according to the vector \( \boldsymbol{b} = (0,2) \) with a probability of \( \frac{1}{3} \). Let \( P_{n} \) be the probability that \( M \) reaches the point \( (0, n) \). Find \( P_{n} \).

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