
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
Does the set of positive numbers \(X = \mathbb{R}_{+}\), with the operations of "addition" and "multiplication by a number" defined as follows for all \(a, b \in \mathbb{R}_{+}\) and \(\alpha \in \mathbb{R}\):

$$
a \oplus b = a \cdot b, \quad \alpha \odot a = a^{\alpha}
$$

form a vector space?

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