
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  $c_1,c_2,\ldots,c_{6030}$  be 6030 real numbers. Suppose that for any 6030 real numbers  $a_1,a_2,\ldots,a_{6030}$ , there exist 6030 real numbers  $\{b_1,b_2,\ldots,b_{6030}\}$  such that \[a_n = \sum_{k=1}^{n} b_{\gcd(k,n)}\] and \[b_n = \sum_{d\mid n} c_d a_{n/d}\] for  $n=1,2,\ldots,6030$ . Find  $c_{6030}$ .

*Victor Wang.*

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