
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 that $a$ is a positive real number, $n$ is a natural number, the parabola $y=-x^{2}+\frac{a^{n}}{2}$ intersects the positive semi-axis of $x$ at point $A$. Let $f(n)$ be the $y$-intercept of the tangent line to the parabola at point $A$.

(1) Express $f(n)$ in terms of $a$ and $n$;

(2) Find the minimum value of $a$ such that $\frac{f(n)-1}{f(n)+1}\geqslant \frac{n}{n+1}$ holds for all $n$;

(3) Compare $\frac{1}{f(1)-f(2)}+\frac{1}{f(2)-f(4)}+\cdots +\frac{1}{f(n)-f(2n)}$ and $6\cdot \frac{f(1)-f(n+1)}{f(0)-f(1)}$ when $0 < a < 1$, and explain the reason.

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