
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
Find the solutions to the following problems:
1. Determine $a$ and $b$ so that $x^{4}+ax^{2}-bx+2$ is divisible by $x^{2}+3x+2$;
2. Given that the quadratic expression in terms of $x$ and $y$, $x^{2}+7xy+ay^{2}-5x-45y-24$, can be factored into the product of two linear factors, find $a$;
3. Given $x+y=1$ and $x^{2}+y^{2}=2$, find the value of $x^{7}+y^{7}$.

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