
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) The two roots of the equation $x^{2}-9x+18=0$ are the base and the leg of an isosceles triangle, respectively. The perimeter of this isosceles triangle is __.

(2) If $(x^{2}+y^{2})^{2}-5(x^{2}+y^{2})-6=0$, then $x^{2}+y^{2}=$ __.

(3) Given that $x$ and $y$ are real numbers and satisfy the equations $x^{2}-2x-6=0$ and $y^{2}-2y-6=0$, respectively, then $\dfrac{x}{y}+ \dfrac{y}{x} =$ __.

(4) If the quadratic equation in $x$, $4x^{2}-4x+c=0$, has two equal real roots, then the value of $c$ is __.

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