
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
Fill in the blanks.

(1) Find the equation of line $l$ that passes through the point $(-2,2)$ and forms a triangle with the coordinate axes having an area of $1$ \_\_\_\_\_\_\_\_\_\_\_\_.

(2) "Algorithmic General" is a rich ancient Chinese mathematics book, which contains the following problem: "Looking at the towering tower with seven layers from afar, the red lights double as they descend, totaling 381 lights. How many lights are on the top of the tower?" This means "A tower has seven layers, from the top to the bottom, the number of lights on each layer is double the number of the layer above. Knowing that there are 381 lights in total on the tower, how many lights are on the top of the tower?" Answer \_\_\_ lights.

(3) Given that the line $y=kx-k+1$ always passes through a fixed point $A$, if point $A$ is on the line $mx+ny-1=0 (mn > 0)$, then the minimum value of $\dfrac{1}{m}+ \dfrac{1}{n}$ is \_\_\_\_\_\_.

(4) In $\Delta ABC$, $a,b,c$ are the sides opposite to angles $A,B,C$ respectively. The correct conclusion number is \_\_\_\_\_\_.

- If $a,b,c$ form an arithmetic sequence, then $B=\dfrac{\pi}{3}$;
- If $c=4,b=2\sqrt{3},B= \dfrac{\pi}{6}$, then $\Delta ABC$ has two solutions;
- If $b=1,ac=2\sqrt{3},B= \dfrac{\pi}{6}$, then $a+c=2+\sqrt{3}$;
- If $(2c-b)\cos A=a\cos B$, then $A=\dfrac{\pi}{6}$.

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