
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
We have  $ 150$  numbers  $ x_1,x_2, \cdots , x_{150}$  each of which is either  $ \sqrt 2 \plus{}1$  or  $ \sqrt 2 \minus{}1$ 

We calculate the following sum:

 $ S\equal{}x_1x_2 \plus{}x_3x_4\plus{} x_5x_6\plus{} \cdots   \plus{} x_{149}x_{150}$ 

Can we choose the  $ 150$  numbers such that  $ S\equal{}121$ ?
And what about  $ S\equal{}111$ ?

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