
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
$(1)$ Given $a$ and $b$ are real numbers.<br/>① If $a+b=13$ and $ab=36$, find $\left(a-b\right)^{2}$;<br/>② If $a^{2}+ab=8$ and $b^{2}+ab=1$, find the values of $a$ and $b$.<br/>$(2)$ If $a$, $b$, $x$, and $y$ satisfy: $ax+by=3$, $ax^{2}+by^{2}=7$, $ax^{3}+by^{3}=16$, $ax^{4}+by^{4}=42$, find the value of $x+y$.

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