
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 functions $f(x)=x+1$ and $g(x)=kx^{2}-x-(k+1)$, where $k>1$. If for any $x_{1}\in [2,4]$, there exists $x_{2}\in [2,4]$ such that $\frac{{f({x_1})}}{{g({x_1})}}=\frac{{g({x_2})}}{{f({x_2})}}$ holds, then the value of real number $k$ is ____.

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