
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
Given a right-angled triangle  $ABC$  and two perpendicular lines  $x$  and  $y$  passing through the vertex  $A$  of its right angle. For an arbitrary point  $X$  on  $x$  define  $y_B$  and  $y_C$  as the reflections of  $y$  about  $XB$  and  $ XC $  respectively. Let  $Y$  be the common point of  $y_b$  and  $y_c$ . Find the locus of  $Y$  (when  $y_b$  and  $y_c$  do not coincide).

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