
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
Find the restriction on the area of the field. Let the area of the field be \( \mathrm{S} \) hectares, then \( 10 \times 300 \times S \leq 10000 \).

\[ \mathrm{S} \leq \frac{10}{3} \]

Let \( 2a = -b \), then \( a = -\frac{b}{2} \).

Consider \( |6|y|+3| \); the value of this expression is obviously not less than three.

Then \( 2|2|x|-b| \leq 9 \), hence \( \mathrm{b} \geq -4.5 \).

For \( b > 4.5 \), the point \( (0, 0) \) does not lie inside the bounded region, which contradicts the logical sense of the problem. Thus, we have a limit from \(-4.5\) to 4.5. Considering that the area increases with the value of \( \mathrm{b} \) and we need only integer values, we have a finite set of parameters: \( \pm 4, \pm 3, \pm 2, \pm 1, 0 \).

Proceeding in ascending order, let \( \mathrm{b} = -4 \):

\[ 2(2x+4) + (6y+3) = 12 \]
\[ x = 0, y = \frac{1}{6} \]
\[ y = 0, x = \frac{1}{4} \]
\[ S = 2 \times \frac{1}{6} \times \frac{1}{4} = \frac{1}{12} < \frac{10}{3}, a = 2 \]

Let \( b = -3 \):

\[ 4x + 6 + 6y + 3 = 12 \]
\[ 4x + 6y = 3 \]
\[ x = 0, y = \frac{1}{2} \]
\[ y = 0, x = \frac{3}{4} \]
\[ S = 2 \times \frac{1}{2} \times \frac{3}{4} = \frac{3}{4} < \frac{10}{3}, a = 1.5 \, \text{ - not an integer} \]

Let \( b = -2 \):

\[ 4x + 4 + 6y + 3 = 12 \]
\[ 4x + 6y = 5 \]
\[ x = 0, y = \frac{5}{6} \]
\[ y = 0, x = \frac{5}{4} \]
\[ S = 2 \times \frac{5}{6} \times \frac{5}{4} = \frac{25}{12} < \frac{10}{3}, a = 1 \]

Let \( b = -1 \):

\[ 4x + 2 + 6y + 3 = 12 \]
\[ 4x + 6y = 7 \]
\[ x = 0, y = \frac{7}{6} \]
\[ y = 0, x = \frac{7}{4} \]
\[ S = 2 \times \frac{7}{6} \times \frac{7}{4} = \frac{49}{12} > \frac{10}{3}; \]
\[ \text{ - not suitable} \]

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