
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 $f(x)=\frac{x^2+2x+a}{x}$, where $x\in[1,+\infty)$, and $a\leqslant 1$.

1. Find the minimum value of the function $f(x)$ when $a=\frac{1}{2}$.
2. Determine the range of values for the real number $a$ such that $f(x) > 0$ always holds for any $x\in[1,+\infty)$.

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