
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
Based on the following set of equations:

$S_1 = 1$;

$S_2 = 2 + 3 = 5$;

$S_3 = 4 + 5 + 6 = 15$;

$S_4 = 7 + 8 + 9 + 10 = 34$;

$S_5 = 11 + 12 + 13 + 14 + 15 = 65$;

$S_6 = 16 + 17 + 18 + 19 + 20 + 21 = 111$;

$S_7 = 22 + 23 + 24 + 25 + 26 + 27 + 28 = 175$;

...

We can find that $S_1 + S_3 + S_5 + \ldots + S_{2n-1} = \_\_\_\_\_\_\_\_$.

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