
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 polynomial $P=2\left(ax-3\right)-3\left(bx+5\right)$, when $x=2$, $P=-31$.<br/>$(1)$ If $a+b=0$, find the values of $a$ and $b$;<br/>$(2)$ Under the condition in $(1)$, if $P \gt 0$, write down the largest integer value of $x$ directly.

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