
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
A gardener plants five maple trees (including three young and two mature maples), six oak trees (including three young and three mature oaks), and six birch trees randomly in a row. What is the probability, in simplified fraction form, that no two birch trees are adjacent to each other? Express your answer as $m+n$ where $\frac{m}{n}$ is the fraction in its lowest terms.

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