
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
Let  $ABC$  be an equilateral triangle with the side of  $20$  units. Amir divides this triangle into  $400$  smaller equilateral triangles with the sides of  $1$  unit. Reza then picks  $4$  of the vertices of these smaller triangles. The vertices lie inside the triangle  $ABC$  and form a parallelogram with sides parallel to the sides of the triangle  $ABC.$  There are exactly  $46$  smaller triangles that have at least one point in common with the sides of this parallelogram. Find all possible values for the area of this parallelogram.

[asy]
unitsize(150);
defaultpen(linewidth(0.7));
int n = 20; /* # of vertical lines, including BC */
pair A = (0,0), B = dir(-30), C = dir(30);
draw(A--B--C--cycle,linewidth(1)); dot(A,UnFill(0)); dot(B,UnFill(0)); dot(C,UnFill(0));
label(" $A$ ",A,W); label(" $C$ ",C,NE); label(" $B$ ",B,SE);
for(int i = 1; i < n; ++i) {
draw((i*A+(n-i)*B)/n--(i*A+(n-i)*C)/n);
draw((i*B+(n-i)*A)/n--(i*B+(n-i)*C)/n);
draw((i*C+(n-i)*A)/n--(i*C+(n-i)*B)/n);
}[/asy]

[Thanks azjps for drawing the diagram.]

<details><summary>Note</summary>*Note:* Vid changed to Amir, and Eva change to Reza!</details>

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