
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 first day, Barry Potter used his magic wand to increase the length of an object by $\frac{1}{2}$. If the initial length was \( x \), it became \( x + \frac{1}{2}x \). On each subsequent day, Barry increased the object's most recent length by $\frac{1}{k+1}$ where \( k \) is the number of the day starting from \( k = 1 \) on day one. Barry continued this pattern, and on the \( n^\text{th} \) day, the object's length became exactly 50 times its original length. Determine \( n \).

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