
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
Given a circle, a line, and points \( A, A', B, B', C, C', M \) lying on this line. According to problems 30.1 and 30.3, there exists a unique projective transformation of this line onto itself, mapping points \( A, B, C \) respectively to \( A', B', C' \). Denote this transformation by \( P \). Construct using only a ruler:
a) the point \( P(M) \);
b) the fixed points of the transformation \( P \) (Steiner's problem).

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