
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
Given two points $O(0,0)$ and $Q(a,b)$, point $P_1$ is the midpoint of segment $OQ$, point $P_2$ is the midpoint of segment $QP_1$, $P_3$ is the midpoint of segment $P_1P_2$, ..., $P_{n+2}$ is the midpoint of segment $P_nP_{n+1}$. Find the limit position of point $P_n$.

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