
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
Let  $O$  be the circumcenter of triangle  $ABC$  with  $|AB|=5$ ,  $|BC|=6$ ,  $|AC|=7$ . Let  $A_1$ ,  $B_1$ ,  $C_1$  be the reflections of  $O$  over the lines  $BC$ ,  $AC$ ,  $AB$ , respectively. Find the distance between  $A$  and the circumcenter of triangle  $A_1B_1C_1$ .

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