
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 a function $y=f(x)$ satisfying $f(x-1) = 2x + 3a$, and $f(a) = 7$.  
(1) Find the expression for the function $f(x)$.  
(2) If $g(x) = x \cdot f(x) + \lambda f(x) + x$ reaches its maximum value of 2 on the interval $[0, 2]$, find the value of the real number $\lambda$.

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