
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
<u>level 1</u>**p1.** Given a circle  $\omega$  and two points  $A$  and  $M$  on  $\omega$ . Take a point  $B$  on  $\omega$  and construct a point  $C$  such that the segment  $AM$  is the median of  $ABC$ . Find  $B$  so that the area of ​​triangle  $ABC$  is maximum.**p2.** A triangle  $ABC$  and a point  $P$  are given. Let  $S_1$  be the symmetric point of  $P$  wrt  $A$ ,  $S_2$  the symmetric point of  $S_1$  wrt  $B$  and  $S_3$  the symmetric point of  $S_2$  wrt  $C$ . Find the locus of  $S_3$  as  $P$  moves.**p3.** Given a segment  $AB$  and a point  $X$  on  $AB$ . The line  $ r$ , perpendicular to  $AB$ , is drawn at  $X$ . Let  $O$  be a point on the line  $ r$  and  $\omega$  the circle with center  $O$  passing through  $A$ . The tangents to  $\omega$   are traced through  $B$ , touching  $\omega$   at  $T_1$  and  $T_2$ . Find the locus of  $T_1$  and  $T_2$  as  $O$  moves on  $ r$ .

<u>level 2</u>**p4.**Given a circle  $\omega$  and a chord  $DA$  on the circle. Take another point  $B$  on  $\omega$  (different from  $A$  and  $D$ ) and draw the line  $s$  passing through  $A$  perpendicular on  $DB$ . The intersection point of line  $s$  and line  $DB$  is  $C$ . Find point  $B$  so that the area of ​​triangle  $ABC$  is maximum.**p5.** Given a triangle  $ABC$ , let  $P$  be any point in the plane. The perpendicular bisectors of  $AP$ ,  $BP$  and  $CP$  that intersect at three points,  $A'$ ,  $B'$  and  $C'$ , are drawn. Find a point  $P$  so that the triangle  $ABC$  and  $A'B'C'$  are similar.**p6.** Let  $C_1$  and  $C_2$  be two intersecting circles at  $A$  and  $B$  where  $AB$  is the diameter of  $C_1$ . Let  $P$  be a point on  $C_2$ , in the interior of  $C_1$ . Construct two points  $C$  and  $D$  on  $C_1$  such that  $CD$  is perpendicular to  $AB$  and  $\angle CPD$  is right, using only the constructions of: $\bullet$  a line given by two points $\bullet$  intersection of  $2$  objects  $\bullet$  line perpendicular on a line.

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