
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]

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[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

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\( \mathbb{R}_0^+ \) is the set of non-negative reals. For \( \alpha \geq 0 \), \( C_\alpha \) is the set of continuous functions \( f : [0, 1] \rightarrow \mathbb{R}_0^+ \) such that:

1. \( f \) is convex \[ f(\lambda x + \mu y) \leq \lambda f(x) + \mu f(y) \text{ for } \lambda, \mu \geq 0 \text{ with } \lambda + \mu = 1 \];
2. \( f \) is increasing;
3. \( f(1) - 2f(\frac{2}{3}) + f(\frac{1}{3}) \geq \alpha \left( f(\frac{2}{3}) - 2f(\frac{1}{3}) + f(0) \right) \).

For which \( \alpha \) is \( C_\alpha \) closed under pointwise multiplication?

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