
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
Determine the real number \( a_{0} \) such that the sequence \(\{a_{n}\}\) defined by the recurrence relation \( a_{n+1} = -3a_{n} + 2^{n} \) (for \( n = 0, 1, \ldots \)) is strictly increasing, i.e., \( a_{n} < a_{n+1} \) for all \( n \geq 0 \).

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