
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  $ABC$  be triangle such that  $|AB| = 5$ ,  $|BC| = 9$  and  $|AC| = 8$ . The angle bisector of  $\widehat{BCA}$  meets  $BA$  at  $X$  and the angle bisector of  $\widehat{CAB}$  meets  $BC$  at  $Y$ . Let  $Z$  be the intersection of lines  $XY$  and  $AC$ . What is  $|AZ|$ ? $ 
\textbf{a)}\ \sqrt{104}
\qquad\textbf{b)}\ \sqrt{145}
\qquad\textbf{c)}\ \sqrt{89}
\qquad\textbf{d)}\ 9
\qquad\textbf{e)}\ 10
$ 

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