
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
Observe the following equations: $S\_1=1$, $S\_2=2+3=5$, $S\_3=4+5+6=15$, $S\_4=7+8+9+10=34$, $S\_5=11+12+13+14+15=65$. A student conjectures that $S\_{2n-1}=(2n-1)(an^2+bn+c)\ (n\in\mathbb{N}^*)$. If the student's answer is correct, then $3a+b=$______.

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