
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) $\sin 75^{\circ}=$ _______

(2) If the two real roots of the equation $x^{2}+(m-1)x+m^{2}-2=0$ are such that one is less than $-1$ and the other is greater than $1$, then the range of values for the real number $m$ is _______

(3) The graph of the function $f(x)=\log_{a}(x+3)-1$ $(a > 0, a \neq 1)$ always passes through a fixed point $A$. If point $A$ lies on the line $mx+ny+1=0$, where $m > 0$, $n > 0$, then the minimum value of $\dfrac{1}{m}+ \dfrac{2}{n}$ is _______

(4) "Solve the equation $\left( \dfrac{3}{5}\right)^{x}+\left( \dfrac{4}{5}\right)^{x}=1$" has the following solution idea: Let $f(x)=\left( \dfrac{3}{5}\right)^{x}+\left( \dfrac{4}{5}\right)^{x}$, then $f(x)$ is monotonically decreasing on $\mathbb{R}$, and $f(2)=1$, so the original equation has a unique solution $x=2$. Following the above solution idea, the solution set of the equation $x^{6}+x^{2}=(x+2)^{3}+(x+2)$ is _______

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