
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 that $a > 0$, proposition $p$: $∀x > 0$, $x + \frac{a}{x} \geqslant 2$ always holds true, and proposition $q$: $∀k \in R$, the line $kx - y + 2 = 0$ intersects with the ellipse $x^2 + \frac{y^2}{a^2} = 1$, find the range of real number $a$ such that $p \vee q$ is true and $p \wedge q$ is false.

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