
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 an isosceles triangle \(ABC\) with vertex angle \(C\) equal to \(100^\circ\), two rays are constructed: one starting at point \(A\) with an angle of \(30^\circ\) to ray \(AB\), and another one starting at point \(B\) with an angle of \(20^\circ\) to ray \(BA\). The constructed rays intersect at point \(M\) inside triangle \(ABC\). Find the angles \(ACM\) and \(BCM\).

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