
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 $f(x)=\frac{1}{x}+a\ln x$, where $a\in\mathbb{R}$:
1. Find the value of the real number $a$ such that the function $f(x)$ takes an extreme value at $x=1$.
2. Under the conclusion of part 1, find the value of $t$ such that the inequality $f(x+1)>\frac{x^2+(t+2)x+t+2}{x^2+3x+2}$ holds for all $x\geq 1$, where $t\in\mathbb{N}^*$.

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