
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 function $f(x)$ with the domain $\{x|x\neq k\pi, k\in \mathbb{Z}\}$, and for any $x, y$ within the domain, the following equation holds: $f(x-y) = \frac{f(x)\cdot f(y)+1}{f(y)-f(x)}$. It is also given that $f(a)=1$ (where $a$ is a positive constant), and $f(x) > 0$ for $0 < x < 2a$.  
(1) Determine the parity of $f(x)$.  
(2) Find the minimum and maximum values of $f(x)$ on the interval $[2a, 3a]$.

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