
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 \(\alpha = \frac{3 - \sqrt{5}}{2}\) and \(f(n) = [\alpha n]\), let us define
\[ F(k) = \min \{n \in \mathbb{N} ; f^{(k)}(n) > 0\} \]
for any \(k \in \mathbb{N}\), where \(f^{(k)}\) represents the \(k\)-th composition of \(f\) with itself. Find the expression for \(F(k)\).

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