
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
A robot is placed at point  $P$  on the  $x$ -axis but different from  $(0,0)$  and  $(1,0)$  and can only move along the axis either to the left or to the right. Two players play the following game. Player  $A$  gives a distance and  $B$  gives a direction and the robot will move the indicated distance along the indicated direction. Player  $A$  aims to move the robot to either  $(0,0)$  or  $(1,0)$ . Player  $B$ 's aim is to stop  $A$  from achieving his aim. For which  $P$  can  $A$  win?

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