
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 quadratic function \( f(x) = x^{2} - tx + t \) where \( t \in \mathbb{R} \), it satisfies the following conditions:
1. The inequality \( f(x) \leq 0 \) has a solution set that contains exactly one element.
2. If \( 0 < x_{2} < x_{1} \), then \( f(x_{2}) < f(x_{1}) \) always holds.

Let \( \{a_{n}\} \) be a sequence with the sum of its first \( n \) terms denoted by \( S_{n} = f(n) \).

(1) Find the general term formula for the sequence \( \{a_{n}\} \).

(2) Compute \( \lim_{n \rightarrow \infty} \left( \frac{1}{a_{1}a_{3}} + \frac{1}{a_{2}a_{4}} + \frac{1}{a_{3}a_{5}} + \cdots + \frac{1}{a_{n}a_{n+2}} \right) \).

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