
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 diagram, point \( C \) is on \( BD \), \(\triangle ABC\) is right-angled at \( B \), \(\triangle ACE\) is right-angled at \( C \), and \(\triangle CDE\) is right-angled at \( D \). Also, \( AB = 2t \), \( BD = DE = 9t \), and \( BC : CD = 2 : 1 \). If the area of \(\triangle ACE\) is \( k \), what is the value of \(\frac{1}{36} k \)?

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