
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
i) How many integers  $n$  are there such that  $n$  is divisible by  $9$  and  $n+1$  is divisible by  $25$ ?
ii) How many integers  $n$  are there such that  $n$  is divisible by  $21$  and  $n+1$  is divisible by  $165$ ?
iii) How many integers  $n$  are there such that  $n$  is divisible by  $9, n + 1$  is divisible by  $25$ , and  $n + 2$  is divisible by  $4$ ?


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