
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
Let $\alpha$ be the remainder of $2^{1000}$ divided by 13, determine the value of $\alpha$.

Determine the value of $\beta=\frac{(7+4 \sqrt{\alpha})^{\frac{1}{2}}-(7-4 \sqrt{\alpha})^{\frac{1}{2}}}{\sqrt{\alpha}}$.

If $f(a)=a-\beta$ and $F(a, b)=b^{2}+a$, determine the value of $\gamma=F(3, f(4))$.

If $\delta$ is the product of all real roots of $x^{\log _{7} x}=10$, determine the value of $\delta$.

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