
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
1. If \(|x+\sqrt{5}|+|y-\sqrt{5}|+|z|=0\), determine \(\alpha=x^{2}+y^{2}+z^{2}\).
2. If \(\beta\) is the sum of all digits of the product \(\underbrace{11111 \ldots 11}_{\alpha \text{ ones}} \times \underbrace{99999 \ldots 99}_{\alpha \text{ nines}}\), determine the value of \(\beta\).
3. Suppose that the real function \(f(x)\) satisfies \(f(xy)=f(x)f(y)\) for all real numbers \(x\) and \(y\), and \(f(1)<1\). Determine the value of \(\gamma=f(\beta)+100-\beta\).
4. If \(n\) is a positive integer and \(f(n)=2^{n}+2^{n-1}+2^{n-2}+\ldots+2^{2}+2^{1}+1\), determine the value of \(\delta=f(\gamma)\).

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