
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
Observe the algebraic expressions and their corresponding values in the table below and answer the questions:

| $x$ | $\ldots $ | $-2$ | $-1$ | $0$ | $1$ | $2$ | $\ldots $ |
|-----|-----------|------|------|-----|-----|-----|-----------|
| $2x-7$ | $\ldots $ | $-11$ | $-9$ | $-7$ | $a$ | $-3$ | $\ldots $ |
| $3x+2$ | $\ldots $ | $-4$ | $-1$ | $2$ | $5$ | $8$ | $\ldots $ |
| $-2x+5$ | $\ldots $ | $9$ | $7$ | $5$ | $3$ | $1$ | $\ldots $ |
| $-3x-1$ | $\ldots $ | $b$ | $2$ | $-1$ | $-4$ | $-7$ | $\ldots $ |

(1) [Initial Perception] According to the information in the table, $a=$______, $b=$______; when $x=$______, the value of $3x+2$ is $7$ greater than the value of $-2x+5$.

(2) [Inductive Rule] The change pattern of the value of $3x+2$ in the table is: for each increase of $1$ in the value of $x$, the value of $3x+2$ increases by $3$; the change pattern of the value of $-2x+5$ is: for each increase of $1$ in the value of $x$, the value of $-2x+5$ decreases by $2$. Similarly, the change pattern of the value of $2x-7$ is: for each increase of $1$ in the value of $x$, the value of $2x-7$ ________; the change pattern of the value of $-3x-1$ is: for each increase of $1$ in the value of $x$, the value of $-3x-1$ ________;

(3) [Problem Solving] If the algebraic expression about $x$ is $mx+n$, and the value of $mx+n$ decreases by $4$ for each increase of $1$ in the value of $x$, and when $x=3$, the value of $mx+n$ is $-5$, find the value of this algebraic expression when $x=7$.

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