
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
In scalene  $\triangle ABC$ ,  $I$  is the incenter,  $I_a$  is the  $A$ -excenter,  $D$  is the midpoint of arc  $BC$  of the circumcircle of  $ABC$  not containing  $A$ , and  $M$  is the midpoint of side  $BC$ . Extend ray  $IM$  past  $M$  to point  $P$  such that  $IM = MP$ . Let  $Q$  be the intersection of  $DP$  and  $MI_a$ , and  $R$  be the point on the line  $MI_a$  such that  $AR\parallel DP$ . Given that  $\frac{AI_a}{AI}=9$ , the ratio  $\frac{QM} {RI_a}$  can be expressed in the form  $\frac{m}{n}$  for two relatively prime positive integers  $m,n$ . Compute  $m+n$ .

*Ray Li.*

<details><summary>Clarifications</summary>[list=1][*]"Arc  $BC$  of the circumcircle" means "the arc with endpoints  $B$  and  $C$  not containing  $A$ ".[/list]</details>

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