
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
"S" is the region of space defined by the inequalities \(x, y, z \geq 0\), \(x + y + z \leq 11\), \(2x + 4y + 3z \leq 36\), and \(2x + 3z \leq 24\). Find the number of vertices and edges of \(S\). For which values of \(a\) and \(b\) is \(ax + by + z \leq 2a + 5b + 4\) satisfied for all points of \(S\)?

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