
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
Given the set \( M = \{ (a, b) \mid a \leq -1, b \leq m \} \). If for any \((a, b) \in M\), it always holds that \(a \cdot 2^b - b - 3a \geq 0\), then the maximum value of the real number \( m \) is _____.

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