
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
\[ f(x) = ax \sqrt{x - 1} + bx + c, \]
where \( x \in [1, +\infty) \), \( a, b, c \in \mathbb{R} \), and \( a \neq 0 \), and the condition \( 0 < 2f(5) = 3f(10) = 4f(17) \leq 1 \), find the maximum value of the real number \( a \).

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