
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
(1) In the expansion of $(1+x)^n$, the coefficient of $x^3$ is 7 times the coefficient of $x$. Find $n$;  
(2) In the expansion of $(ax+1)^7$ ($a\neq 0$), the coefficient of $x^3$ is the arithmetic mean of the coefficients of $x^2$ and $x^4$. Find $a$;  
(3) It is known that in the expansion of $(2x+x^{\log x})^8$, the value of the term with the largest binomial coefficient is 1120. Find $x$.

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