
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
(The full score for this question is 12 points) Given the function $f(x) = x^2 + 2x + a$, where $x \in [1, +\infty)$.

(1) When $a = $, find the minimum value of the function $f(x)$;

(2) If for any $x \in [1, +\infty)$, $f(x) > 0$ always holds, try to find the range of the real number $a$.

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