
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 set of equations:

$S_{1}=1$,

$S_{2}=2+3+4=9$,

$S_{3}=3+4+5+6+7=25$,

$S_{4}=4+5+6+7+8+9+10=49$,

...

Based on the equations above, guess that $S_{2n-1}=(4n-3)(an+b)$, then $a^{2}+b^{2}=$_______.

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