
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
The problem is reduced to finding the common PPC (Production Possibility Curve) for three individual functions:

$\mathrm{M}=40-2 \mathrm{~K}$, $\mathrm{M}=64-\mathrm{K}^{\wedge} 2$, $\mathrm{M}=64-\mathrm{K}^{\wedge} 2$.

The task can be simplified by combining the PPC of two males from the previous point, $\mathrm{M}=128-0.5 \mathrm{~K}^{\wedge} 2$, with the PPC of one female. Males have the PPC $\mathrm{M}=128-0.5\mathrm{~K}^{\wedge} 2$ and the female has the PPC $\mathrm{M}=40-2\mathrm{K}$. The opportunity costs for the males are $A_I=\mathrm{K}_1$ and for the female, $-2$.

We have:

$\mathrm{K} \leq 2$: $\mathrm{K}_1 = \mathrm{K}$, $\mathrm{K}_2 = 0$

$2<\mathrm{K}\leq 22$: $\mathrm{K}_1 = 2$, $\mathrm{K}_2 = \mathrm{K} - 2$

$22 < \mathrm{K} \leq 36$: $\mathrm{K}_1 = \mathrm{K} - 20$, $\mathrm{K}_2 = 20$

Substituting these values into the function $\mathrm{M}=128-0.5 \mathrm{~K}_1 \wedge 2+40-2 \mathrm{~K}_2$, we get the combined PPC of the male and female:

$\mathrm{M}=168-0.5 \mathrm{~K}^{\wedge} 2$, $\mathrm{K} \leq 2$

$\mathrm{M}=170-2 \mathrm{~K}$, $2<\mathrm{K} \leq 22$

$\mathrm{M}=20 \mathrm{~K} - 0.5 \mathrm{~K} \wedge 2 - 72$, $22<\mathrm{K} \leq 36$

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