
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 defined by:
\[ y \geqslant 0, \]
\[ y \leqslant x, \]
\[ y \leqslant 2 - x. \]
Region \( N \) changes with \( t \) and is defined by the inequalities:
\[ t \leqslant x \leqslant t + 1, \]
with \( t \) ranging from \( 0 \) to \( 1 \). Find the function \( f(t) \) that represents the area of the intersection of regions \( M \) and \( N \).

\[
f(t) = \qquad
\]

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