
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
The numbers in the sequence 101, 104, 109, 116,  $\dots$  are of the form  $a_n = 100 + n^2$ , where  $n = 1$ , 2, 3,  $\dots$ .  For each  $n$ , let  $d_n$  be the greatest common divisor of  $a_n$  and  $a_{n + 1}$ .  Find the maximum value of  $d_n$  as  $n$  ranges through the positive integers.

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