
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
Given that the vertex of angle $α$ is at the origin, its initial side coincides with the positive semi-axis of the $x$-axis, and its terminal side passes through point $P(-3, \sqrt{3} )$.
(1) Find the value of $\tan α$;
(2) Define the determinant operation $\begin{vmatrix}a & b \\ c & d\end{vmatrix} =ad-bc$, find the value of the determinant $\begin{vmatrix}\sin α & \tan α \\ 1 & \cos α\end{vmatrix}$;
(3) If function $f(x)=\begin{vmatrix}\cos \left(x+α\right) & -\sin α \\\sin \left(x+α\right) & \cos α\end{vmatrix} \left(x\in \left[ -\dfrac{\pi }{4},\dfrac{\pi }{4} \right]\right)$, find the maximum value of the function $y= \sqrt{3}f\left( \dfrac{π}{2}-2x\right)+2{f}^{2}\left(x\right)$, and indicate the value of $x$ when the maximum value is reached.

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