
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
Let  $\phi(n)$  denote the number of positive integers less than or equal to  $n$  which are relatively prime to  $n$ . Over all integers  $1\le n \le 100$ , find the maximum value of  $\phi(n^2+2n)-\phi(n^2)$ .

*Proposed by Vincent Huang*

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