
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 consider two circles  $k_1(M_1, r_1)$  and  $k_2(M_2, r_2)$  with  $z = M_1M_2 > r_1+r_2$  and a common outer tangent with the tangent points  $P_1$  and  $P2$  (that is, they lie on the same side of the connecting line  $M_1M_2$ ). We now change the radii so that their sum is  $r_1+r_2 = c$  remains constant. What set of points does the midpoint of the tangent segment  $P_1P_2$   run through,  when  $r_1$  varies from  $0$  to  $c$ ?

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