
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
Points  $P$ ,  $Q$ , and  $M$  lie on a circle  $\omega$  such that  $M$  is the midpoint of minor arc  $PQ$  and  $MP=MQ=3$ . Point  $X$  varies on major arc  $PQ$ ,  $MX$  meets segment  $PQ$  at  $R$ , the line through  $R$  perpendicular to  $MX$  meets minor arc  $PQ$  at  $S$ ,  $MS$  meets line  $PQ$  at  $T$ . If  $TX=5$  when  $MS$  is minimized, what is the minimum value of  $MS$ ?

*2019 CCA Math Bonanza Team Round #9*

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