
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
Let the trigonometric function \( f(x) = \sin \left( \frac{k x}{5} + \frac{\pi}{3} \right) \) where \( k \neq 0 \).

1. Find the maximum value \( M \), minimum value \( m \), and the minimum positive period \( T \) of \( f(x) \).
2. Determine the smallest positive integer \( k \) such that for any variation of the variable \( x \) between any two integers (including the integers themselves), the function \( f(x) \) has at least one value that is \( M \) and another value that is \( m \).

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