
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
Two circles touch each other and the sides of triangle \(ABC\). The first circle, with radius \(\frac{1}{18}\), touches the sides \(AB\) and \(AC\) at points \(L\) and \(K\), respectively. The second circle, with radius \(\frac{2}{9}\), touches the sides \(AC\) and \(BC\) at points \(N\) and \(M\), respectively. Find the area of the triangle \(ABC\) if \(AL = \frac{1}{9}\) and \(CM = \frac{1}{6}\).

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