
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
In the coordinate plane, there are two regions $M$ and $N$. Region $M$ is determined by the inequalities $y \geqslant 0, y \leqslant x$, and $y \leqslant 2-x$. Region $N$ changes with $t$ and is determined by the inequality $t \leqslant x \leqslant t+1$, where $0 \leqslant t \leqslant 1$. Determine the function $f(t)$ representing the common area of regions $M$ and $N$.

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