
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
A line that is neither horizontal nor vertical intersects the $y$-axis when $x=0$ and intersects the $x$-axis when $y=0$. For example, the line with equation $4 x + 5 y = 40$ intersects the $y$-axis at $(0,8)$ and the $x$-axis at $(10,0)$.

(a) The line with equation $2 x + 3 y = 12$ intersects the $y$-axis at $A$ and the $x$-axis at $B$. If $O$ is the origin, determine the area of $\triangle AOB$.

(b) Suppose that $c > 0$. The line with equation $6 x + 5 y = c$ intersects the $y-axis at $D$ and the $x$-axis at $E$. If $O$ is the origin and the area of $\triangle DOE$ is 240, determine the value of $c$.

(c) Suppose that $m$ and $n$ are integers with $100 \leq m$ and $m < n$. The line with equation $(2m)x + y = 4m$ intersects the $y$-axis at $P$ and the $x$-axis at $Q$. The line with equation $(7n)x + 4 y = 28n$ intersects the $y$-axis at $S$ and the $x$-axis at $R$. If quadrilateral $PQRS` has area 2022, determine two possible pairs $(m, n)$.

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