
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
A finite arithmetic progression \(a_{1}, a_{2}, \ldots, a_{n}\) with a positive common difference has a sum of all its terms equal to \(S\), and \(a_{1} > 0\). It is known that if the common difference of the progression is increased 4 times while keeping the first term unchanged, then the sum \(S\) increases by 3 times. By how many times will \(S\) increase if the common difference of the original progression is doubled (keeping the first term unchanged)?

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