
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
How many functions \( f: \{0,1\}^{3} \rightarrow \{0,1\} \) satisfy the property that, for all ordered triples \(\left(a_{1}, a_{2}, a_{3}\right)\) and \(\left(b_{1}, b_{2}, b_{3}\right)\) such that \( a_{i} \geq b_{i} \) for all \( i \), \( f\left(a_{1}, a_{2}, a_{3}\right) \geq f\left(b_{1}, b_{2}, b_{3}\right) \)?

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