
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 $P$ be a highest point on the graph of the function $f(x) = \sin(\pi x)$, and $Q$ be a lowest point on the graph of the function $g(x) = \cos(\pi x)$.
(1) Find the minimum value of $|PQ|$;
(2) Find the area of the triangle formed by three adjacent intersection points of the graphs of $f(x)$ and $g(x)$;
(3) Find the analytical expression of the function $h(x)$ whose graph is symmetric to the graph of $f(x)$ about the line $x = \frac{1}{4}$, and find the range of $h(x)$ for $x\in[-\frac{2}{3}, \frac{1}{3}]$.

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