
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
Find the number of 16-tuples \(\left(x_{1}, x_{2}, \ldots, x_{16}\right)\) such that:
(i) \(x_{i} = \pm 1\) for \(i = 1, \ldots, 16\),
(ii) \(0 \leq x_{1} + x_{2} + \cdots + x_{r} < 4\), for \(r = 1, 2, \ldots, 15\),
(iii) \(x_{1} + x_{2} + \cdots + x_{10} = 4\).

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