
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
Let \( x \) and \( y \) be the lengths of two segments on the left side. The length of the third segment is \( L-(x+y) \). The sample space consists of all pairs \((x, y)\) for which 

\[ 
\begin{cases}
0 \leq x \leq L \\
0 \leq y \leq L \\
0 \leq L - (x + y) \leq L 
\end{cases} 
\Leftrightarrow 
\begin{cases}
0 \leq x \leq L \\
0 \leq y \leq L \\
0 \leq x+y \leq L 
\end{cases}
\]

In the coordinate plane, these points form a triangle with vertices \((0,0)\), \((L, 0)\), and \((0, L)\).

The area of this triangle is \( L^2 / 2 \).

To solve the problem, it is convenient to compute the probability of unfavorable events, defined by the system of inequalities

\[
\begin{cases}
x \leq \frac{5}{12} L \\
y \leq \frac{5}{12} L \\
L - (x + y) \leq \frac{5}{12} L 
\end{cases}
\Leftrightarrow
\begin{cases}
x \leq \frac{5}{12} L \\
y \leq \frac{5}{12} L \\
x + y \geq \frac{7}{12} L 
\end{cases}
\]

In the coordinate plane, these points also form a right triangle with legs \( L / 4 \).

The area of this triangle is \( \frac{1}{32} L^2 \). The probability of unfavorable events is the ratio of the areas

\[
q = \frac{\frac{1}{32} L^2}{\frac{1}{2} L^2} = \frac{1}{16}
\]

The desired probability is

\[
p = 1 - q = 1 - \frac{1}{16} = \frac{15}{16}
\]

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