
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
Find a real number  $t$  such that for any set of 120 points  $P_1, \ldots P_{120}$  on the boundary of a unit square, there exists a point  $Q$  on this boundary with  $|P_1Q| + |P_2Q| + \cdots + |P_{120}Q| = t$ .

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