
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
Consider all the real sequences  $x_0,x_1,\cdots,x_{100}$  satisfying the following two requirements:
(1) $x_0=0$ ;
(2)For any integer  $i,1\leq i\leq 100$ ,we have  $1\leq x_i-x_{i-1}\leq 2$ .
Find the greatest positive integer  $k\leq 100$ ,so that for any sequence  $x_0,x_1,\cdots,x_{100}$  like this,we have
\[x_k+x_{k+1}+\cdots+x_{100}\geq x_0+x_1+\cdots+x_{k-1}.\]

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