
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
A point $M$ lies inside a triangle. Let $k_{1}, k_{2}, k_{3}$ be the distances from $M$ to the sides of the triangle and $h_{1}, h_{2}, h_{3}$ be the corresponding heights. Find the minimum value of the expression

$$
\left(\frac{k_{1}}{h_{1}}\right)^{\alpha}+\left(\frac{k_{2}}{h_{2}}\right)^{\alpha}+\left(\frac{k_{3}}{h_{3}}\right)^{\alpha} \quad \text { for } \alpha \geqslant 1
$$

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