
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
The table shows a set of equations, and based on it, a student conjectured that $S_{2n-1} = (2n-1)(an^2 + bn + c)$. The teacher confirmed that the conjecture was correct. Determine the value of $a - b + c$.
The equations are given as follows:
$$
S_1 = 1, \\
S_2 = 2 + 3 = 5, \\
S_3 = 4 + 5 + 6 = 15, \\
S_4 = 7 + 8 + 9 + 10 = 34, \\
S_5 = 11 + 12 + 13 + 14 + 15 = 65, \\
\ldots
$$

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