
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
An integer-valued function $f$ is called intense if $f(x) + f(y) + f(z) > z^3$ for all positive integers $x, y, z$ with $x \neq z$ and $y \neq z$. Let $h$ be an intense function such that $h(1) + h(2) + \dots + h(15)$ is as small as possible. Compute the minimum possible value for $h(10).$

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