
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 the table as a set of equations, a student conjectures that $S_{2n-1}=(2n-1)(an^{2}+bn+c)$. The teacher confirms the conjecture is correct. Then, $a-b+c=$ \_\_\_\_\_.
- $S_{1}=1$,
- $S_{2}=2+3=5$,
- $S_{3}=4+5+6=15$,
- $S_{4}=7+8+9+10=34$,
- $S_{5}=11+12+13+14+15=65$,
- $…$

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