
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 function $f(x) = e^x + ax$, the equation of the tangent line to the curve $y = f(x)$ at the point $(0, f(0))$ is $y = 1$.  
(1) Determine the value of the real number $a$ and the intervals where the function $f(x)$ is monotonic;  
(2) If $b > 0$, and $f(x) \geq b(b-1)x + c$, find the maximum value of $b^2c$.

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