
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 a rectangular coordinate plane, given points \( P_{1}(1,2), P_{2}\left(2, 2^{2}\right), P_{3}\left(3, 2^{3}\right), \cdots, P_{n}\left(n, 2^{n}\right) \), where \( n \) is a positive integer. For any point \( A_{0} \) on the plane, let \( A_{1} \) be the point symmetric to \( A_{0} \) with respect to point \( P_{1} \), \( A_{2} \) be the point symmetric to \( A_{1} \) with respect to point \( P_{2} \), ..., \( A_{n} \) be the point symmetric to \( A_{n-1} \) with respect to point \( P_{n} \).

1. Find the coordinates of the vector \(\overrightarrow{A_{0} A_{2}}\).
2. When point \( A_{0} \) moves along the curve \( C \), the trajectory of point \( A_{2} \) is the graph of the function \( y = f(x) \), where \( f(x) \) is a periodic function with a period of 3, and when \( x \in (0, 3] \), \( f(x) = \log x \). Determine the analytic expression of the function that has the curve \( C \) as its graph for \( x \in (1, 4] \).

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