
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, and based on the equations above, guess that ${{S}_{2n-1}}=\left( 4n-3 \right)\left( an+b \right)$, then ${{a}^{2}}+{{b}^{2}}=$ ____.

${{S}_{1}}=1$

${{S}_{2}}=2+3+4=9$,

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

${{S}_{4}}=4+5+6+7+8+9+10=49$, (......)

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