
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
In the Cartesian coordinate system, it is known that the coordinates of point $P_0$ are $(1,0)$. Rotate $P_0$ around the origin in a counterclockwise direction by 45° to get $P_1$, extend $0P_1$ to point $P_2$ such that $0P_2 = 20P_1$; then rotate $P_2$ around the origin $0$ in a counterclockwise direction by 45° to get point $P_3$, extend $OP_3$ to point $P_4$ such that $0P_4 = 2OP_3$… Continue in this manner, then the coordinates of point $P_{2008}$ are.

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