
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 points $Q_1, Q_2, \ldots, Q_9$ on a straight line, in the order stated (not necessarily evenly spaced). Let $Q_3$ and $Q_7$ each be the center of a cluster of three closely spaced points. Find the point such that the sum of the undirected lengths $QQ_1, QQ_2, \ldots, QQ_9$ is minimized.

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