
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
Let  $p\in \mathbb P,p>3$ . Calcute:

a) $S=\sum_{k=1}^{\frac{p-1}{2}} \left[\frac{2k^2}{p}\right]-2 \cdot \left[\frac{k^2}{p}\right]$   if  $ p\equiv 1 \mod 4$ 

b)  $T=\sum_{k=1}^{\frac{p-1}{2}} \left[\frac{k^2}{p}\right]$    if  $p\equiv 1 \mod 8$ 

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