
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 the sets \( M = \{1, 2, 3\} \) and \( N = \{1, 2, 3, 4, 5\} \), define the function \( f: M \rightarrow N \). Let the points \( A(1, f(1)), B(2, f(2)), C(3, f(3)) \) form a triangle \( \triangle ABC \). The circumcenter of \( \triangle ABC \) is \( D \), and it is given that \( \mu DA + DC = \lambda DB (\lambda \in \mathbb{R}) \). Find the number of functions \( f(x) \) that satisfy this condition.

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