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

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