
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
On a unit square grid, points $A$, $B$, $C$, and $D$ are the vertices of a $10 \times 10$ square. There are pieces at points $A$, $B$, and $C$. These pieces are moved by a sequence of steps to the points $B$, $A$, and $D$ respectively; in each step, one of the pieces is moved to an adjacent grid point, but they never leave the square. At every position, the area of the triangle formed by the pieces is at least $t$. What is the largest possible value of $t$?

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