
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
Given an arithmetic sequence \(\{a_n\}\) with the first term \(a_1=1\) and the common difference \(d > 0\), where its second, fifth, and fourteenth terms are the corresponding second, third, and fourth terms of a geometric sequence \(\{b_n\}\).  
(1) Find the general formula for the sequences \(\{a_n\}\) and \(\{b_n\}\);  
(2) Suppose there is a sequence \(\{c_n\}\) such that for any positive integer \(n\), \(\dfrac{c_1}{b_1} + \dfrac{c_2}{b_2} + \ldots + \dfrac{c_n}{b_n} = a_{n+1}\) holds. Find the value of \(a_1c_1 + a_2c_2 + \ldots + a_nc_n\).

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