
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
Observe the following equations:

① $ \cos 2\alpha = 2 \cos^2 \alpha - 1 $;

② $ \cos 4\alpha = 8 \cos^4 \alpha - 8 \cos^2 \alpha + 1 $;

③ $ \cos 6\alpha = 32 \cos^6 \alpha - 48 \cos^4 \alpha + 18 \cos^2 \alpha - 1 $;

④ $ \cos 8\alpha = 128 \cos^8 \alpha - 256 \cos^6 \alpha + 160 \cos^4 \alpha - 32 \cos^2 \alpha + 1 $;

⑤ $ \cos 10\alpha = m \cos^{10} \alpha - 1280 \cos^8 \alpha + 1120 \cos^6 \alpha + n \cos^4 \alpha + p \cos^2 \alpha - 1 $;

It is conjectured that $ m - n + p = \_\_\_\_\_\_\_\_ $.

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