
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
Given the hyperbola $4x^2 - y^2 = t$ ($t \neq 0$), its two asymptotes and the line $x = \sqrt{2}$ enclose a triangular area (including the boundary) denoted as $D$. $P(x, y)$ is a point within $D$. The objective is to find the minimum value of the target function $z = \frac{1}{2}x - y$.

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