
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
If $x_{0}, y_{0}, z_{0}$ is a solution to the simultaneous equations below, $\left\{\begin{array}{l}x-y-z=-1 \\ y-x-z=-2 \\ z-x-y=-4\end{array}\right.$

If $\beta$ is the remainder of $\underbrace{111 \cdots 111}_{10011 \text{ 1's}} \div \alpha$, determine the value of $\beta$.

If $\gamma$ is the remainder of $\left[(\beta-2)^{100}+\beta^{50}+(\beta+2)^{25}\right] \div 3$, determine the value of $\gamma$.

If the equation $x^{4}+a x^{2}+b x+\delta=0$ has four real roots with three of them being $1, \gamma$ and $\gamma^{2}$, determine the value of $\delta$.

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