
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 two regions in the Cartesian coordinate plane: 
\[ M = \left\{ (x, y) \in \mathbf{R}^2 \mid 0 \leq y \leq \min(2x, 3-x) \right\} \]
\[ N = \left\{ (x, y) \in \mathbf{R}^2 \mid t \leq x \leq t+2 \right\} \]

Let \( f(t) \) denote the area of the intersection of \( M \) and \( N \). Find the expression for \( f(t) \) when \( 0 \leq t \leq 1 \).

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