
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 the function $f(n) = \log_{n+1}(n+2)$ where $n \in \mathbb{N^*}$, define $k \in \mathbb{N^*}$ such that the product $f(1) \cdot f(2) \cdot \ldots \cdot f(k)$ is an integer as the "yearning number." Then there are ____ yearning numbers within the interval $[1, 50]$.

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