
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
On the initial day, Barry Sotter used his magic wand to make an object's length increase by $\frac{1}{3}$. If the object was originally $x$ meters long, after his magic, it became $x + \frac{1}{3}x$. On each subsequent day, he increased the object's latest length by $\frac{1}{k+2}$ (where $k$ is the day number starting from $0$). Barry continues this daily routine and on the $n^{\text{th}}$ day, the object's length becomes exactly 50 times its original length. Determine the value of $n$.

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