
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 $f(x)$ on the defined interval $[a, b]$ satisfying (1) $f(x) > 0$; (2) $f'(x) < 0$; (3) for any $x_1, x_2 \in [a, b]$, the inequality $\displaystyle f\left(\frac{x_1 + x_2}{2}\right) \leq \frac{f(x_1) + f(x_2)}{2}$ holds. Define $S_1 = \int_{a}^{b} f(x) \, dx$, $S_2 = \frac{f(a) + f(b)}{2} \cdot (b-a)$, and $S_3 = f(b)(b-a)$. Find the relationship between $S_1$, $S_2$, and $S_3$ in ascending order.

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