
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
Define \( f: \mathbb{R}^{2} \rightarrow \mathbb{R}^{2} \) by \( f(x, y) = (2x - y, x + 2y) \). Let \( f^{0}(x, y) = (x, y) \) and, for each \( n \in \mathbb{N} \), \( f^{n}(x, y) = f(f^{n-1}(x, y)) \). Determine the distance between \( f^{2016} \left( \frac{4}{5}, \frac{3}{5} \right) \) and the origin.

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