
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 $xOy$, there is a sequence of points $A_1(x_1, y_1)$, $A_2(x_2, y_2)$, ..., $A_n(x_n, y_n)$, ... that satisfy the following recursive relationship:
$$
\begin{cases}
x_{n+1}= \frac {1}{2}(x_{n}+y_{n}) \\
y_{n+1}= \frac {1}{2}(x_{n}-y_{n})
\end{cases}
$$
Given that $A_1(1,1)$, calculate the limit as $n$ approaches infinity of the sum of the distances from the origin $O$ to each point $A_i$:
$$
\lim_{n\rightarrow \infty }\left(|OA_{1}|+|OA_{2}|+\ldots+|OA_{n}|\right) = \text{?}
$$

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