
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
We know that if a function $f(x)$ defined on an interval satisfies the inequality $$\frac{f(x_1)+f(x_2)}{2} \leq f\left(\frac{x_1+x_2}{2}\right)$$ for any two numbers $x_1$, $x_2$ in that interval, then the function $f(x)$ is called a convex function on that interval. Similarly for a sequence $\{a_n\}$, if it holds that $$\frac{a_n + a_{n+2}}{2} \leq a_{n+1}$$ for any positive integer $n$, then the sequence $\{a_n\}$ is called a convex sequence. Given a sequence $\{a_n\}$ that satisfies the following two conditions:
1. The sequence $\{a_n\}$ is a convex sequence, with $a_1 = 1$ and $a_{10} = 28$;
2. For positive integers $n$ ($1 \leq n < 10$, $n \in \mathbb{N^*}$), it holds that $|a_n - b_n| \leq 20$, where $b_n = n^2 - 6n + 10$;
Determine the range of the third term $a_3$ in the sequence $\{a_n\}$.

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