
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
The quadratic equation in one variable $x^{2}+mx-1=0$, when $m=1$, the positive root of this equation is called the golden ratio. A rectangle whose ratio of width to length is the golden ratio is called a golden rectangle. The design of the Parthenon temple in Greece and the optimal selection method of the famous Chinese mathematician Hua Luogeng also apply the golden ratio.

$(1)$ Find the golden ratio.

$(2)$ Given real numbers $a$ and $b$ satisfy: $a^{2}+ma=1$, $b^{2}-2mb=4$, and $b\neq -2a$, find the value of $ab$.

$(3)$ Given two distinct real numbers $p$ and $q$ satisfy: $p^{2}+np-1=q$, $q^{2}+nq-1=p$, find the value of $pq-n$.

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