
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 $f(x) = x + \frac{a}{x}$ ($a > 0$),

(1) If $f(1) = 2$ and $f(m) = 5$, find the value of $m^2 + m^{-2}$;

(2) Find the range of the real number $a$ such that the function $f(x)$ is increasing on the interval $(1, +\infty)$.

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