
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
In 1778, the French mathematician Lagrange proposed a theorem in the co-authored book "Analytic Function Theory": If the function $y=f\left(x\right)$ satisfies the following two conditions: ① its graph is continuous on the closed interval $\left[a,b\right]$; ② it has a derivative on the interval $\left(a,b\right)$. Then there exists at least one number $c$ in the interval $\left(a,b\right)$ such that $f\left(b\right)-f\left(a\right)={f'}\left(c\right)\left(b-a\right)$, where $c$ is called the Lagrange mean value. The Lagrange mean value $c$ of the function $g\left(x\right)=\ln x+x$ on the interval $\left[1,2\right]$ is ____.

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