
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 day  $1$  of the new year, John Adams and Samuel Adams each drink one gallon of tea. For each positive integer  $n$ , on the  $n$ th day of the year, John drinks  $n$  gallons of tea and Samuel drinks  $n^2$  gallons of tea. After how many days does the combined tea intake of John and Samuel that year first exceed  $900$  gallons?

*Proposed by Aidan Duncan*

<details><summary>Solution</summary>*Solution.*  $\boxed{13}$ The total amount that John and Samuel have drank by day  $n$  is   $$ \dfrac{n(n+1)(2n+1)}{6}+\dfrac{n(n+1)}{2}=\dfrac{n(n+1)(n+2)}{3}. $$   
Now, note that ourdesired number of days should be a bit below  $\sqrt[3]{2700}$ . Testing a few values gives  $\boxed{13}$  as our answer.</details>

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