
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 the inequalities:
\[
\begin{cases}
y \geq 0, \\
y \leq x, \\
y \leq 2 - x,
\end{cases}
\]
Region \( N \) is a region that changes with \( t \), and is defined by the inequality:
\[
t \leq x \leq t + 1
\]
where \( t \) ranges from \( 0 \) to \( 1 \). Find the common area of regions \( M \) and \( N \) as a function of \( t \), denoted by \( f(t) \).

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