
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
On the real number line, paint red all points that correspond to integers of the form  $81x+100y$ , where  $x$  and  $y$  are positive integers. Paint the remaining integer point blue. Find a point  $P$  on the line such that, for every integer point  $T$ , the reflection of  $T$  with respect to  $P$  is an integer point of a different colour than  $T$ .

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