
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  $P(x)$  be a polynomial with positive integer coefficients and degree 2015. Given that there exists some  $\omega \in \mathbb{C}$  satisfying  $$ \omega^{73} = 1\quad \text{and} $$    $$ P(\omega^{2015}) + P(\omega^{2015^2}) + P(\omega^{2015^3}) + \ldots + P(\omega^{2015^{72}}) = 0, $$  what is the minimum possible value of  $P(1)$ ?

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